I'd really love to know these things about the abilities that the bottled fairies give:
*How am I supposed to improve the abilities' levels?
*What exactly are them for?
*Is it worth giving abilities to my pets?
1) There are some neggs that increase their levels, or sometimes the Wheel of Excitement does.
2) They all do different things. Some heal, some defend, some freeze. They're only useful in the battledome.
3) If you're going to battle, you should probably get some abilities. If you're not interested in battling, they're not terribly useful.
The In-Depth Battlepedia's got a good run-through of them here: