Why does bible say we all are sinners?

It's important to understand up front that God is a God of LIFE, not death! He is a God with a plan and that plan is to make billions more like Him, who value what He values and want to live and enjoy eternal life with Him.

However, in order for us to 'choose' good and right like He always does, we need to experience the horror and pain and death caused by sin. This age is to live out that experience and eventually make a choice to follow Him and his son Jesus or to follow the devil and his ways of deception, lies, suffering and death.

God's plan is proceeding perfectly.

We are all part of Adam - part of the mistake he made of doing his own thing and disobeying God. We share in his sin and the way of living that thinks first of self. When Adam chose to follow the serpent/devil instead of God, he separated himself and his descendants from knowing God through love and respect and trust. Damage was done to the whole world when that happened and we are all part of that broken relationship. Gen 3 speaks of the trouble that resulted form the first sin. We cannot escape that reality by somehow living a good life.

It's ok to be a sinner, that's the world we live in - the Devil is the god of this world and he makes it the way he wants it. But God's plan to redeem or save all was arranged before the world began. He is not trying to fix it! In Christ, it is already fixed. Rev 13:8

Jesus was always going to be the saviour and change the future from death to all to life for all - for all to be forgiven of sin and trust in his sacrifice made for us.

You have stumbled on a great passage that describes what is going on. It continues in v16

For if by the trespass of the one, the many died, how much more did the grace of God and the gift in grace, which is of the one man Jesus Christ, abound to the many!

just as sin reigned in death, so also grace might reign through righteousness, unto eternal life through Jesus Christ our Lord. v21

We are all sinners because we all sin. None are righteous like God and cannot be on our own. Only in Jesus are we declared righteous and free from sin and the penalty of sin - death.

We don't need to worry about sin - God has already addressed it and provided a perfect solution. Even if we lived life perfectly from now on, we cannot redeem ourselves from past sin and the penalty of death, God decrees, sin deserves. He made us, He gets to make the rules. He knows GOOD and EVIL and He knows that to live with evil is very very sad and hurtful.

The One forming light and creating darkness, Causing well-being and creating disaster; I am the LORD who does all these things. Isaiah 45:7

He doesn't want that for His creation, but He does need for us to experience it so we can make an informed choice and decide to follow Jesus into eternal life and not the devil into death.

You seem to ask why is God punishing us for what Adam did?

He is not punishing in a sense of anger and vengeance. He is allowing the consequences of sin to run their course - death (in this age) is the ultimate penalty.

For the LORD disciplines those he loves, and he punishes each one he accepts as his child.

All are loved by God and He wants all to receive the gracious blessing of salvation in Christ. We are all God's children, though most have no idea and are unaware of the future God has already provided.

GratefulDisciple already provided a useful overview; I am going to address some points more directly.

As far as I know [Jesus'] story, he was a good, compassionate and benevolent man

Jesus is the Christ. He is much more than a (mere) man, even according to Unitarians. According to Trinitarians, He is God. Jesus kept the law perfectly. He wasn't simply good, compassionate and benevolent, He was without sin.

but these statements of all being sinners felt somehow so ill and bad.

Sin is bad; it's not surprising you would have such feelings. Without knowing more, however, it's hard to say if what you experienced was the Holy Spirit whispering that you are a sinner in need of forgiveness, or the Adversary whispering that this is a bunch of nonsense you should ignore. In either case, however, I would suggest it is hopeful that you feel this way. Most Christians will struggle with sin and feelings of inadequacy; this is a sign of the spiritual battle that is being waged for your soul, that will only be won (in your favor) by Christ's second coming. The adversary does not give up on those that might still be dragged down, so if you aren't feeling uncomfortable, you are either unusually blessed with an incredible faith (unlikely, given your question), or the Adversary is convinced he's already won your soul and doesn't need to keep fighting for it.

if Adam or Eve did a mistake, why did god punish humanity for sin and death?

Adam and Eve were created to be Good, and to have a perfect relation with God. When that relation was damaged, they lost that connection (note in Genesis 3:8 that Adam and Eve no longer see God) and gained the capacity (and inclination) to do evil deeds.

Tolkien has an interesting take on the question of death: do you really want to be separated from God for all eternity? In fact, that eternal separation is the most accurate and consistent definition of Hell. Death is, in a sense, a gift; an assurance that our broken condition is not eternal.

It is also a gift to Creation as a whole. The bible teaches that Man's corruption, prior to Noah, grew and grew. You've almost certainly heard the expression "power corrupts". Well, sin corrupts, and a person's capacity to do evil can grow with time. (Consider all the serial killers that started out torturing animals and worked their way toward ever more heinous acts.) Do you really want such a person to live forever, to have all of eternity to "perfect" their evil? Death is a judgement and incarceration that none can evade or escape, preventing evil from growing unchecked. Some have theorized that this is why God shortened human lifespans after the Flood.

Sure, death does have a bigger motive in this world but the statement used to justify this is so not correct or even good.

I'm not sure what motive you ascribe, but seeing as you are a professed non-Christian, I can guess. Historically, Christians did not accept Uniformitarianism and Evolutionism. Although many Christians have been bullied into accepting such "science", others still believe in the bible's clear teaching that all life was created by God in six days, and find this more consistent with the available evidence than Uniformitarianism and Evolutionism. The only "constructive" purposes of death are as I describe in the previous paragraph.

(I could go on at length on this subject, but we're getting off topic. Instead, I will say only this: Naturalism is not science. It's a philosophy — a religion by another name — that has an inherently anti-Christian agenda. Anyone desiring to be Christian would do well to bear this in mind when evaluating whether or not to believe "scientific" claims which are founded on Naturalist dogma, rather than simply accepting such claims on the basis of "consensus".)

Also, how can we justify that Christ died of (sic) 'our' sins? How can our generations blame for that?

Our generation is sinful, just like the ones before and the ones that will follow, until the Second Coming.

I think your doubt is because you are asking the wrong question. Yes, in a pedantic sense, Christ died "because of" our sins, but it is much more correct to say that He died for our sins. Because of His great love for us — for all people* — He chose to bear the burden of all sins, for all people, for all time. Had He wanted to, He could have avoided death, but then we would be judged by our own, insufficient works. By choosing to bear the punishment for sin — even yours and mine — we can stand before God, justified by Christ's atonement.

This is not a "free pass". We are still called to faith and repentance, and to try to avoid egregiously continuing in sin. We can still reject the gift God freely offers. But it is also a gift. Nothing we can do can overcome our sin and "earn" salvation.

Shouldn't his sacrifice be seen and future generations make sure that no innocent or a good person / any life form out of our survival zone dies in his honour and for the love he showered on our ancestors?

Creation is still subject to the Curse, and will be until the Second Coming. As I recently explained elsewhere, there are no "good" people. While we might wish that all would look to Christ's example and lead better lives as a result (indeed, one might argue this is even a goal of Christianity), it is clear that evil still exists. (Why God continues to allow it to exist is a question worthy of many books, and starts to touch on matters of free will.)

God's standard is perfection. If you have ever used foul language, you are sinful; you are neither "good" nor "innocent". If you have ever thought ill of another person (or animal), or looked at someone other than your spouse with impure thoughts... if you have ever failed to love God with all your heart and mind, or failed to love your fellow humans as yourself, then you are neither "good" nor "innocent".

Hopefully that addresses the question in your subject, as well as some of the other points you raised.

Because your question is an overview / introductory question, I will provide a high level overview answer to emphasize the connection among several issues you raised. Although in Christianity there is no standard explanation for each issue (and the discussion can easily take a book each), there is enough agreement in all mainstream Christian denominations. I will limit my answer to generalities plus some contemporary (but orthodox) theories that explain the issues to a 21st century audience.

The reality of sin and the Christian explanation

When a Christian say that we are all sinners, at the bottom is an empirical observation of our tendency to cause damage, suffering, and destruction in animals, nature, other people, and even ourselves ! This FACT is empirical, requiring no religion to teach.

The second observation requiring no religion to teach is that even the most well-brought up children with minimal evil influence from their social environment will grow up to have this tendency. No one teach them to be selfish, but as they grow up they will find within themselves this conflict where they want to do good and be loving, but yet something is holding them back.

But Christianity teaches that a good God created a good world intended for good human beings to manage and develop through flourishing good lives in loving relationship with this God. How does it come to this?

The teaching is called Original Sin, where Adam and Eve came in, which is related to the common terminology you mentioned "born into sin". The doctrine provides an explanation how Adam's rebellion against God, for some reason, caused human nature itself to be corrupted, producing the evil tendency mentioned above. I understand if you feel the doctrine is inadequate to explain it; you are far from alone. One resource I highly recommend is an interview video of a Christian philosopher Eleonore Stump: Why Is Sin?. After beginning with the empirical description of the fallen human nature, she started talking about original sin at 4:22 and tried to explain it with analogy from the recent (but controversial) movie Noah (2014) of how the whole human species is God's child that He cannot abandon since it's his creation.

God's "human nature repair project" through Jesus

If you are a father who cannot abandon his child who as a species rebels against him, what does it take for this conflicted child to be healed so it can function the way the child has been meant to be before the original sin?

This is the project of Salvation. Another interview video with Elonore Stump How Does Salvation Work emphasizes that it's CRITICAL to understand what we are saved FROM: that there's something wrong with us, as if humanity has contracted an HIV virus that could break out into AIDS if not treated. She then describes our part to receive this salvation: not trying to be good or to be a hero in fixing ourselves, but to surrender to rock-bottom, to "not resisting" to allow God to heal us, just as we don't resist getting vaccinated.

Christianity teaches that species-wide problem through one man's Original Sin requires species-wide solution through one man's Grace (free gift). This is exactly what Romans 5:12-21 is talking about, the comparison between Adam and Jesus, who in Christian theology has the role of Second/Last Adam.

Who is Jesus? Christianity teaches that Jesus is God himself, the same God who created the human species, who literally TAKES ON a 100% human nature to be born into this world as one of us and therefore can retain his 100% divine nature so Jesus can operate through both. How this works is explained by Eleonore Stump in another interview video Jesus as God using the analogy of God putting on a very dark contact lens for a role in a movie as a blind man (because human nature is limited).

How exactly does Jesus's sacrifice on the cross provides salvation? Again, it's a very complex topic that theologians name the Atonement. Christians are free to choose any Biblical theory as long as they accept that Christ saves them.

• Eleonore Stump in an interview video How Does Atonement Work explains a theory in terms of dealing with our shame, guilt, and parts of us that we don't want others not to see, let alone God ! The second problem is a "fragmentation of our will" problem: that we do want to be loved but at the same time we don't want to be loved. This is the conflict she explained in the Sin and the Salvation videos. Her theory is how God himself in Christ's passion, naked, tortured, rejected, violated, shamed in front of his own family (and mother!) with his arms out saying to us "come to me" helps us to surrender to God's love for us. In this image, we don't see the God who is judging and condemning but we see God who sees us as so precious that He is willing to die for us. He values and desires us that much, and this removes our shame.
• N.T. Wright, a New Testament scholar, explains in same interview series of another theory in terms of 1) letting evil runs its course concentrating its power in a vortex and lets it be exhausted, 2) a sequel of self-giving love which created the world in the first place, and 3) finish the plan of renewed creation where the original vocation that God gave humans to do can be fulfilled