Is there regret in Heaven?

Reflecting on Ray Butterworth's answer, this verse has a second possible interpretation:

And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away. — Revelation 21:4

Is the wiping away of tears instantaneous? Or does it take time?

If the wiping away of tears takes time, that includes a period during which regrets are processed and addressed, leading eventually to a blissful state.

God is a comforter, not a brainwasher.

most Christians believe in some form of eternal salvation.

Seeing as this is a fundamental tenant of Christianity, I think it would be safe to say "all". Note John 3:16, "For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life." (Emphasis added.)

While there are some groups with, shall we say, more exotic beliefs, the majority of mainstream Christians would likely classify such beliefs as heretical, i.e. said groups are not Christian by the standards of mainstream Christianity.

(Side note: many people use "heaven" to refer to this afterlife. Strictly speaking, however, "heaven" is a physical location which exists now, in our current reality, and not one necessarily suited to human habitation. Indeed, "heaven" and "outer space" may be the same. Believers, in eternity, will inhabit the New Creation, which is a new Earth that has not been tainted by the Fall and the Curse. Keep this in mind any time you think you've found some contradiction regarding the nature and/or location of "heaven".)

And in this eternally saved state, the soul of a person will be free from all worry, regret, sadness, etc.

This is less clear. It's true that Revelation 21:4 says that "[God] will wipe away every tear from their eyes", and that "neither shall there be mourning, nor crying, nor pain"... but in order for God to wipe away tears, there need to be tears. Will no one ever stub their toe in the New Creation (TNC)?

Personally, I tend to agree with the view I've seen expressed elsewhere that TNC epitomizes the expression "time heals all wounds", and that Revelation 21:4 is more a statement of mean state than absolute state. That is, someone in TNC might still experience pain or sadness, but that these states will be unusual and (at least in comparison to eternity) highly transitory.

Conversely, there's something to be said for the argument that this is why pain and suffering exist in this life; because they give us valuable perspective that won't be available in TNC.

Alice loves Bob, and worries that Bob is not saved. Bob dies without salvation. This makes Alice sad. Alice dies and goes on to her eternal reward. Does she still care about the fact that Bob never reached salvation? Will she carry that sadness for eternity, or does she cease to care about Bob?

Initially, she might, but eternity is a long time to hold on to a regret.

If she doesn't carry that sadness for eternity, but she still cares about Bob's eternal soul, in what sense does she care if the fact that he never achieved salvation causes her no worry or pain? If she knows that during the vast majority of her eternal existence, she won't care about Bob's eternal soul, why does she bother to care about it during her life?

Christians are called to be loving and compassionate, both of which require at least caring about the fate of others. (Most would argue it requires some effort to affect said fate.) Our time on this Earth is the time in which we have the potential to influence others. Once Bob dies, his eternal fate is set¹, but while he is alive, it is loving to try to win him to Christ. Ultimately, it's the difference between trying to do something and failing, versus a willful decision to not try at all. Such a willful decision is contrary to the Loving nature of God which Christians are called to emulate, even sociopathic, and it's hard to imagine a genuine Christian holding such an attitude. Genuine faith is fruitful (James 2:14-26). This is not to say that works are salvific (Ephesians 2:8-9), but rather works are an indicator of salvation. Thus, Alice should care because God cares, and if Alice doesn't care, that would be serious cause to question whether Alice has faith.

(¹ Contrary to Ray's answer, as far as I know most Christians believe that this life is the only chance one gets to accept God, and that those who don't are condemned. Just what condemnation entails is subject to significant debate, though it's generally agreed that it means something other than eternal life with God. This is in contrast to universalism, which believes that all people will eventually be "restored" somehow. While attractive, I personally can't see how universalism can be reconciled with the many, many places that the Bible warns of 'alternatives to Salvation'.)

No one will feel regret:

And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away. — Revelation 21:4

Bob might have died without salvation, but it's very unlikely that he really understood what it meant or that he knowingly rejected it.

God desires that all receive salvation, but he is not currently attempting to convert the world (if he were, he's doing a very poor job of it).

The fact is that this age isn't the only time of salvation. In fact, very few people of this age will ever receive salvation (consider the billions of people that spent their lives without ever even hearing of Jesus).

It's in the second general resurrection, at the end of the Millennium, that the vast majority of mankind will receive their first real chance of salvation. Bob will almost certainly be among those resurrected to a second physical life here on Earth, and will almost certainly understand and accept salvation at that time.

See my answer to What is the Order of the Resurrection of the Dead?