The main reason cacti (or most plants) won't flower is insufficient light. Think of it's natural habitat. Cacti love to bake out there in the sun all day. It looks as though it's been begging for more light for quite a while. Yes, you can give a trim and a bigger pot, but more importantly, it want's more light. Try to find a south or west facing window where it can get full on sun.

There's not lot you can do right away to the plant as it is. You can easily take cuttings though and start new plants, which would be the best thing to do as you will not be able to rescue this one and make it a decent shape.. Simply cut off a piece the size you want, and allow the cut surface to dry. Then insert it into a dry gritty compost and wait!
Any stems that have had the upper part removed may well regrow but most likely with several new growths coming from the aureoles (the places where the spines grow from) so it wouldn't really help to tidy the plant much.

The pot is far too small, which is probably why the growth is spindly and weak. There is nothing to be done with the bent stems - you might as well cut them off and discard them. Any further help you want please ask - you can send me a PM if you like.