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.