To be honest I shouldn't be the one answering your questions but I just can't watch people to drown before me.

-If I'm this bad at 20 why go on?
First of all you should explain what do you consider "this bad" as. And explain it to yourself. You might not be that bad at all. I know this might seem a little bit narcissist but think about worse people than you. That might cheer you up. But at that time of course your depression might bring up people who are better than you. Compare yourself to those who are better than you and think... Why are they better than me? And on what aspect? What is being better and why the hell should I consider them better than me? Is wealth a sign of being better? Is it being successful? Answer these questions first.
-Is it because I think my problems are more important than others?
Indeed your problems are more important than others at least it should be and at least for you. Do not bother yourself that much with society but do NOT force yourself away from it too for a person is weak but people are strong. You won't be able to stand alone and no one can. But place yourself above the others for if you are not nothing else matters.
-If nobody is happy why bother?
Are you happy? I presume your answer to this question will be "no" for you are suffering under depression and I will say one thing. You have 2 things to struggle and live for. First is your own happiness and the second is if you love someone it is their happiness. Actually to be honest the second is just a form of the first for the happiness of the loved ones cause you to be happy. Well the mechanics of the brain aren't that important as long as you are happy.
-If they are just whiny people who can't deal with minor distress, aren't I too?
Actually I will take some risk and... Yes, you are too. But look that on the bright side, if you are just a whiny person who can't deal with minor distress it means your problems are minor. If your problems aren't minor than you are not like them. Either way you deserve to be happy.

I hope my answers satisfy you. Have a good day...