I still have a few doubts. Thank you very much for helping me.

1) I'm writing some letters at the moment.
But: I'm going to write some letters (future) .
2) I spend a lot of time (a long time/much time) doing homework. (Which are possible?)
3) He set off on a journey in search for gold.
4) Did you manage to do the exercise? (But not: Could you finish the exercise?/Have you managed to do the exercise?)
5) I'd like to be a climate champion because I think that young people are the future and that all together we can make it. I can't stand who (?) damage the world.

All are fine until the very last sentence.

I can't stand people who ...