Do you need to start a new game after patches?

No. The only reason to do so is to get the full benefit, but other than that there's no reason to do so.

It varies really.

Small issues and stuff you haven't run into yet, will generally be fixed in an existing game.

Issues you've already encountered, quests that have already glitched won't always be fixed by the patch as that info is stored in your save game and patches won't touch those. These ones will require either a new game start or reloading a save from before you encountered the bug.

This somewhat controversial. DT explained it rather well. I, personally start a new game, with every patch release. I run several mods, I feel the game is more stable when starting with a fresh clean slate. Mods or not, if I am experiencing less issues than others, then it may or may not be due to this. But it makes me feel better.

So start a new game now or wait for 1.8 (PS3)?

You don't have to. I'm still with the character I started on 11/11/11 and no problems (except capturing Winterhold for the legion).

When is that coming out? I would not let the "potential" release of a possible patch, hinder me from playing. Not at all.