As far as I am aware, according to Christian beliefs after Adam's fall all humanity became sinful and doomed to perdition. Then Son of God was sent by Father to save mankind and give eternal live for those believing in Him.
‌Why was all this needed? Why didnt God simply send a prophet with instructions how people can save themselves? Why did God have a need to make sacrifice for himself?
First disclaimer: I hope you're not looking for this to be a quick, glib, bumper-sticker synopsis of theology, because it doesn't really work that way.
Second disclaimer: I suspect the complete answer to this question involves knowledge about the universe and the nature of God that we're not privy to at this time.
Now, having said that, I think there is a lot of nuance to Christianity, issues and questions that theologians have debated for centuries, that you're pretty much glossing over or missing entirely. Let me give you my take on it. As CS Lewis said about his explanations, if it helps you, run with it. If not, feel free to discard it entirely.
As I see it, what Adam's fall really did was awaken mankind to the sinful side of his own nature, and the down side to free will, ie. the ability and desire to choose to disobey God, and thus move ourselves farther away from Him spiritually. The problem, of course, is the eventual, ultimate end of that scenario. Now, before you jump to conclusions as to what the end is, let me explain my take on THAT.
In "The Screwtape Letters", CS Lewis said that in the end, everything in the universe will either be claimed by God or by Satan, God on the "pedantic" grounds that He created it, or Satan on the more "dynamic" grounds of conquest. Now, admittedly, Lewis was speaking from the perspective of a demon for the purposes of the book's premise, so the viewpoint is a bit skewed by that, but the way I see it, when we die, our souls go to and become part of one side or the other. Think of it, if you will, as Buddhism without reincarnation, sort of a Nirvana and Anti-Nirvana.
Okay, now the choices that you make in life with that free will are going to move you closer to one side or the other. But it's not as simple as just "I did something wrong, I'm sorry, forgive me", because those choices leave a mark on your soul, they change who you are. And they create a . . . oh, a sort of magnetism that draws you toward more of the same kind of choices. The more you make decisions to behave as a good person, the more you BECOME a good person, and the easier it is for you to choose that path the next time a decision has to be made; likewise, the more you allow yourself to behave badly, the easier it becomes to make the wrong choice the next time.
It takes something kinda big and metaphysical to wipe away the marks on your soul made by the bad decisions you've made. Sometimes I think of it in the same light as making a New Year's resolution: you announce your decision to make a change (even if you're announcing it just to yourself) because it makes it more real, more solid, more committed. I'm pretty sure there's more to it in this case, but it's that general idea. Prior to Jesus' birth as a human, the Israelites signified that commitment to abandoning disobedience and cleansing away the dark stains on their souls by sacrificing an animal, sort of a smaller death to symbolize the larger death that would be the ultimate end of continuing the path of dark and disobedient choices.
Ultimately, though, this isn't how God wanted it to be, the relationship between us and Him hemmed in by the constant need for ritual and strained by the constant pulling of our darker nature away from Him. So Jesus agreed to come to Earth, live as a human, die as a human, and be resurrected as the deity He truly was, a sort of "one time pays for all" sacrifice to take the place of all those smaller, stopgap sacrifices, one Ultimate death for Him to replace all the ultimate deaths for us. Basically, the cumulative bad choices we have all made, or all will make, would have made us part of the dark, the Anti-Nirvana, the property of Satan, however you want to think of it. We doomed ourselves in eternity through our choices in life. Instead, Jesus took that doom, the consequences of our choices, the stains on our souls, onto Himself.
Fundamentalist Christian sects teach that when He died, he actually did go to Hell, just as we would absent the sacrifice to wipe away the marks on our souls. But because He was, in His spiritual essence, God and not a regular human, He was able to win Himself free and be resurrected and returned to Heaven. Go with it or don't, as you please. But then end result is that the consequences of our choices have now been paid forward for us.