[DKP on it to check its current points , Killing a mob does give it exp but wont help it evolve that's for its actual level , which is base 30 and maxed when you get it , ED can raise its cap of levels to max 45 so you can work on its stats and such , to evolve, it need to hit something , so yes training eles are the way to go , it takes about 10-12hr is to max its form , while it also maxes its skills.
You'll have to macro a spell on it to raise its Resisting spells. I for one macro a feeding every 30min when I afk train it for that long , but that's up to you. You don't have to feed it , every successful command raises its loyalty , but every unsuccessful lowers it , so get tame/lore as high as you can. You could macro a command with the spell (bless so it fights faster) to keep it happy for 10-12hrs.
Some come breedable but they don't soposed to be. So either page for it to get "fixed" buy a deed for 10K tokens , or just simply don't do it. You can be jailed and both pets taken if caught.