1) Don't really know about skinning, but mining is PLENTY profitable at pretty much all levels. That said, two gathering professions would get you more gold than one.
2) If you're planning to actually make use of the items engineering can give you, then the time to switch is now. Engineering is fairly easy to level up if you keep at it, especially since as a miner you can provide yourself with most of the materials you'll need to craft engineering items. If you just want the high-level goodies, then waiting until you're 70 and are sitting on a pile of gold will make powerlevelling it a lot easier.
3) Engineering isn't really a class-related profession. It has some goodies for almost everyone, as opposed to, say, blacksmithing, which is more for melee types like warriors. In my opinion, it's the best profession to have if you plan on doing PvP a lot, because the bombs and things you can make will be very helpful if you use them properly. Also, the goblin jumper cables are nice for when you're in a party and wipe. You can just feign death, then res your healer and have them res everyone else. Of course, that assumes the blasted things will work...
In the end though, engineering is still the joke profession. It's crap at making money (you can basically make a few quest items, scopes and fishing lures) but it has loads of cool toys and, of course, pets. I <3 my mechanical yeti :)