Does a fighters power develop when moving up weight divisions or not?

Idk lol. For some people yes, others no. There's lots of examples of fighters who have discontinued draining themselves and found greater power higher up, then probably an equal amount who've seen their power diminish

 

Yes, more weight behind the punch => harder. But it is usually at the expense of a bit of speed.

 

In an absolute sense yes. But relatively speaking no. Because the opponents in that higher weight class will on average also have more power and more punch resistance.