Questions 1 thru 6 would take an historical scholar to answer. However, I want to contribute to an answer to both questions 6 and 7. First question 6, in my experience and understanding, cause and effect are important to understand because it leads one to the understanding of the four noble truths, and specifically, to the understanding of dukkha. I'm not trying to be cryptic here, only that to fully explain this would take a very long explanation. Now question 7, the Buddha taught rebirth because that is true whereas reincarnation is not true. There is nothing that continues into another incarnation. Rebirth is the re-entry by the deathless (that's what the Buddha called it in his first teaching of the five ascetics) into another living experience. So it's a fresh start, not a reincarnation. The karma brought into the new experience is simply the accompanying ignorance. As one awakens, karma vanishes, in exact accordance to the self-awareness (which, of course, is not a "self").

1. This is out of scope for Buddhism SE. This is more like a history question. The Buddhist texts describe Charvaka as being materialist. Sarva-Darsana-Samgraha describes Charvaka as being both materialist and hedonist, but some scholars (especially Bhattacharya) say there's insufficient evidence to claim that Charvaka was hedonist.

2. The explanation of the Buddhist position given in the Sarva-Darsana-Samgraha doesn't match what's in the Pali Canon, but rather appears to expand on it. So, I guess this may describe the position of Indian Buddhist scholars who lived long after the Buddha's time.

I will skip #3 and #4.

5. From my understanding, Charvaka was not the most popular school at the time. Brahmanism (the Vedic religion) was the most popular orthodox religious view of the era. Charvaka was a minority heterodox school.

6. We don't know.

And lastly question 7 will be answered below.

First mark of existence

Sabbe sankhara anicca - All conditioned things/phenomena are impermanent.

Apart from Nibbana, everything else is conditioned. Something conditioned depends on something else, is influenced by something else, and is caused by something else. And all such things are impermanent, changing, unreliable, unstable.

So, from this, you can see that causation is quite important.

Everything that exists and happens is caused by something else. Events that happen to us and our existence, may not all be caused by karma, but they are definitely caused by something, be it weather, the actions of others, bile, etc. according to the Sivaka Sutta.

Conditioned phenomena are essentially chains of dependent processes.

Second mark of existence

Sabbe sankhara dukkha - All conditioned things/phenomena are suffering.

Now, the term "conditioned things" is focused on mind-body phenomena. Wherever there's mind-body phenomena occurring, craving arises, and that's the cause of suffering (second noble truth).

Also, take note that it is suffering that is occurring due to craving. It's not your suffering or my suffering. It's not your craving or my craving. It's just suffering caused by craving.

Third mark of existence

Sabbe dhamma anatta - All things/phenomena are not self.

"All things" include Nibbana.

This is the teaching that truly differentiates what the Buddha taught from what everybody else taught, including Charvaka.

The view of Charvaka is that there is a self. It's associated with the physical body. When the physical body is destroyed, the self would be destroyed too.

The Buddhist view is that the self and being are just emergent phenomena. A being is a term applied conventionally when the five aggregates occur together and when there is the clinging of the mind to the idea of "I am".

The self is just the mental idea of "I am". The self idea arises from moment to moment, from the mind-body phenomena, depending on conditions. It's always changing, just as the five aggregates are always changing.

If you step into a flowing river then step out then later step in again, would you be stepping into the same river twice? It may appear so, but in reality, the molecules, flow speed, temperature, pressure etc. of the river would have changed. So, it's not the exact same river.

Similarly, the mental idea of the self keeps changing throughout one's life, and even throughout a single day. Also consciousness, the mind, the body, is also changing. The mind changes more rapidly than the body.

Thinking that there is a concrete self, a specific individual identity, a core entity, that is born, lives, grows old, dies, and then is reborn or perhaps not reborn, are all self-views. The Charvaka view is called annihilationism because they believed that there was a self (associated with the physical body) and then it will become destroyed (along with the physical body).

According to the Vina Sutta, when you break down the five aggregates of form, feeling, perception, consciousness and mental formations down to their constituent parts, you cannot find the self anywhere. This is explained using the analogy of the lute. When you break the lute down to its constituent parts, you cannot find music anywhere.

So, all phenomena is not self. All phenomena is empty of a self (Sunna Sutta).

Karma

The Charvaka view is that if you will become destroyed at death, why bother with good behavior and charity? This is typical of a self-view. If suffering will end at death, why bother trying to end suffering? Just endure it till it's gone.

The Buddhist view is that firstly there's causation, as described above. Karma is one of the causes of suffering but is not the only cause, according to Sivaka Sutta.

The Buddhist goal is to end suffering, not in the future, but in the here-and-now. If we talk about ending suffering in the future, then that future is described relative to the self. That's a self-view.

Here's an example of self-view related to the future. A person thinks he will probably live for another 50 years hopefully, if he has good health. And he thinks, "perhaps it's good if I can attain Nibbana within my natural life, before death comes, otherwise I may be reborn somewhere else and not fortunate enough to encounter the Dhamma in another life". That's self-view.

Ending of suffering in Buddhism is through the purification of the mind.

Hereafter and Rebirth

Charvaka says there's no rebirth and no hereafter. Everything ends at death. What ends? The self ends. This is a self-view. That there was a self, and it got destroyed at death with no continuity.

What does Buddhism say?

According to Buddhism, there is causation. Every event and every object is caused by something else. It is influenced by something else. It depends on something else.

According to Buddhism, there is continuity. Everywhere the mind-body phenomena occurs, you have individuality or the mental idea of the self reappearing, and with that, suffering also reappears. Individuality is (re)born. The self idea is (re)born. Suffering is (re)born.

But this causation, continuity, individuality, suffering, craving etc. are all not self. They represent chains of dependent processes.

And what is "not self"? It means there is no core entity, no concrete self, no specific individual identity that is born, lives, dies and is reborn or not reborn. That's just a temporary emergent phenomena that arises from the mind and body.

Also please see MN 38 (story of monk Sati), SN 22.85 (what happens to the Tathagata after death?), SN 12.20 (have I existed in the past? will I exist in future?), SN 12.17 (is suffering self-made or others-made, or both or neither?).

Fictional contrivance? Or skillful means?

The teaching given to people that they would be reborn depending on their karma, is skillful means (upaya) on the part of the Buddha, given in AN 5.57. In fact, there are multiple self-views given in that sutta like "I will grow old" etc. As explained in AN 5.57, the Buddha taught these reflections for a reason.

From AN 5.57:

“And for the sake of what benefit should a woman or a man, a householder or one gone forth, often reflect thus: ‘I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do’? People engage in misconduct by body, speech, and mind. But when one often reflects upon this theme, such misconduct is either completely abandoned or diminished. It is for the sake of this benefit that a woman or a man, a householder or one gone forth, should often reflect thus: ‘I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do.’

If people who have self-view reflect that their actions will result in their future condition, then misconduct will be completely abandoned or diminished.

This is not a fictional contrivance. It's skillful means (upaya). Perhaps you can call it a "mind trick".

Remember that everything conditioned are chains of dependent processes? The Buddha used this reflection to change the conduct of people with self-view in order to reduce suffering and lead towards the complete elimination of suffering.

Skillful means occurs elsewhere too like in AN 7.52. Giving in charity with expectation of future good results in return is the lesser purpose, while adorning the mind or purifying the mind is the highest purpose. For someone with self-view, charity with the lesser purpose is better than no charity at all, and is a step in the right direction.