Yes, as we all suspected, witches are Puella Magi gone bad. It's interesting, I've been re-watching the beginning of this series with my sister and at the beginning Kyubey says to them something to the effect of, "If Puella Magi are made from wishes, then witches are made from curses." Very cool to see that play out here where Sayaka effectively cursing herself sped up her transformation.

As for Homura, perhaps her wish was to control time. Offensive abilities as a Puella Magi seem to be separate, and randomly assigned to specific girls. Though as we saw in Sayaka's case, the type of wish granted also affects abilities. So perhaps Homura just wished to go back in time to before Kyubey contacted Madoka, and the time-manipulation abilities were a side-effect of the wish she made.