Impermanence, nonexistance of self, meaning of life I've been learning lately about Buddhism and I have several questions about it. I haven't actually read any of the sacred books, just some stuff on the internet and now I'm reading What Makes You Not a Buddhist as an intro to Buddhism, so I might have some misconceptions about it. 

So let's get to the questions:

1) In What Makes You Not a Buddhist the author says that asking a Buddhist about the meaning of life is inappropriate because it implies that somewhere there's a final purpose of being. Do Buddhists believe that there's no purpose in living? And if there is, what is it based upon. Where is the motivation to do anything if it isn't building up to something? Why be good/smart/compassionate/... ? And why not commit a suicide instead of learning and understanding things so that you don't suffer - wouldn't that be faster?

2) If I understand correctly Buddhism says that we don't exist, we're just made up of different parts and our conception of ourselves is just an illusion. But if Descartes was right when he said "I think, therefore I am.", then I don't see how that works with Buddhism - someone here must be wrong (I think). Do Buddhists believe that there's no essence that is "me" (like soul in Christianity), just random parts connected together in a certain point of time? If yes, then what is that thing that thought of the illusion of existing(Is it our mind? Does it also not exist?)?

3) One of the four seals is: All compounded things are impermanent. In the book author says that all things are compounded so saying "All things are impermanent" is equivalent. How is that even possible if anything we understand about the world is true? Everything is made out of atoms and atoms are made out of subatomic particles and perhaps those are made out of sub subatomic particles and so on (I don't know), but in the end, there must be a basic building block that makes those things and that's bound to be constant. It's hard for me to understand how everything could be a compounded because that means that it's made out of smaller things put together. But if those things are also compounded, this process goes on infinitely. But there's no infinity in nature.

I'll really appreciate the answers.

EDIT: Thank you all for the great answers. They really helped me understand some things better.
Because I hate deciding on which answer is the best, I'll let the community do it - after some time I'll pick the one with most upvotes.
Do Buddhists believe that there's no purpose in living?

No.Buddhists believe the purpose in living is to free one's self from suffering.

And if there is, what is it based upon?

The universal need for happiness.

Where is the motivation to do anything if it isn't building up to something?

Now that you know the Buddhist goal is to be free from suffering this is where all the motivation goes to. Buddhism teaches the Eight Noble Path that when practiced,cultivated,developed and perfected will lead to happiness.

Why be good/smart/compassionate/... ?

Because choosing to do good is better than choosing to do bad.Also Buddhists believe in karma so the more bad things you do the more suffering follows.

And why not commit a suicide instead of learning and understanding things so that you don't suffer - wouldn't that be faster?

Because Buddhists believe if you commit suicide you'll just get reborn but the problems stays the same, if not worse.Not only is it pointless but by committing suicide you are creating a lot of bad energy.More bad karma than you can handle even.As killing is a very heavy karma and that includes killing yourself.You may also acquire the habit or tendency to commit suicide in future lives and trap your self in a never ending circle of suffering.

This is why it is Better to understand things so you don't suffer.But even if you can't understand anything.It's still better to Not commit suicide.Because that's will save you enormous amount of suffering.

Do Buddhists believe that there's no essence that is "me" (like soul in Christianity), just random parts connected together in a certain point of time? 

Yes no Essence (like soul in Christianity).

If yes, then what is that thing that thought of the illusion of existing(Is it our mind? Does it also not exist?)?

It's one of the random parts connected together in a certain point of time you mentioned before. Though it's not random it follows a pattern of cause and effect.In other words it's conditioned.

"All things are impermanent" is equivalent. How is that even possible if anything we understand about the world is true? 

I don't understand.What do you see in the world that makes you think everything is permanent.People are born and die.Food are bought and rots.Leaves turn green and falls.Flowers bloom and wither.2 years olds eventually become 80 years olds.You were thinking of jogging now your thinking of what's on tv.You were cold now your warm.You were standing now your sitting and now your standing up again.You had the intention to read about Buddhism now you have the intention to write a post.Even an atom is always moving.Always.Nothing stays fixed the way it is FOREVER.A tv might look a tv now but in 100 years it's probably molten liquid.

One of the reason people mistaken things for being permanent is because of not being able to see things from a wider time scale.