On Draenor population it says "full", does that mean I may not be able to log onto characters now and then or how am I supposed to read that?

There may be queues at the start of new expansion (it never happens, for example, now in february), in exchange you get market that is not dead, big number of guilds. Pick any of top 10 horde-populated servers (see population size on realmpop or wowprogress)

I'm not actually sure if I know how to apply to a guild. Is it correct I just have to press "J" and request on as many as possible and hope? Is there a smarter way?

Browse guilds on wowprogress, you can sort them by language, server, faction, number of raiding days per week; in-game guild finder is good only for leveling alts to get guild bonuses.