Happiness points will only be gained in battles in which experience is earned; thus only appositely in NPC/wild Pokemon battles, not PvP battles.
To clarify for the level-up ambiguity: when your Pokemon reaches a new level, it will auto-gain 5 additive happiness points.
Lastly, it's not just Eevee's evolutionary chain that is evolvable via happiness; as of now,
all happiness-based evolutions should be evolvable at this time, barring some branched (multi-evo-line) evolutions.
For those multi-line evolutions that are fully evolvable, they have been hard-coded, meaning that the evolutions must be implemented manually. It is intended that all evolutions will be supported, however—Shane admitted that some may have been omitted, so if you notice any multi-line evolutions that aren't evolvable, as they were in the handheld games, feel free to report it, and it'll be imparted to Shane, or any other Admin/DEV, who can implement it.
DeGe.DF said:
poliwhirl can evolve to palpitoed? :v
So far, all item-evolvable Pokemon with multi-line evolutions have been hard-coded in, such as Poliwhirl, so it should indeed be evolvable into Politoed via the King's Rock now.