You're correct as soon as you reach level 50 you can use all the gear you have from any other characters.

You can spend your CP early but won't gain more until you're level 50.

Gear for PvP comes from various sources depending on your build, not every set from PvP works best in PvP and quite often sets from PvE dungeons will be superior. I would however advise you don't make any gear legendary (gold) quality nor invest too much money in buying gear just yet as Morrowind is out in just over a month and will change the games meta quite a bit as to what's considered good.

Like already been said, yes, past lvl 50 your cp level is account wide, though do keep in mind gear cap is cp 160(not level 50 or cp 600). Another thing it means is that once your character hits level 50, provided you already have some cp in your account, he will MASSIVELY outlevel his old gear because as far as the game is concerned he won't be lvl 50 in lvl 45 gear but lvl cp 160(for example, using this number as it's the gear cap atm) in lvl 45 gear which will make all of his stats drop dramatically. Don't let that discourage you and just hurry up to upgrade your armor

And yes, unfortunately a lot of competitive PvP gear, most of it even I'd say, is acquired through PvE. That includes some open world sets that are farmable on your own or buy-able from other people as they're BoE, some dungeon sets that you'll have to group with other people to acquire(although you can relatively easily solo most normal dungeons and they drop the same sets as vet versions, only in blue quality; but since you can upgrade your gear easily anyway, it only matters for jewelry which you cannot upgrade) and monster sets, the helms for which only drop in vet dungeons off the final boss, and shoulders are to be found by completing daily dungeon quests from the Undaunted. There is also craftable gear which you can just craft/ask someone else to craft, but most of which is not very competitive atm. It all depends on the build you decide to play though.