What should I consider when choosing which monsters to keep?

Monsties gain slightly different stat boosts when you hatch them, but these numbers are small and fixed. Added to these numbers are the base stats, which are identical among all monsties of the same species and level. So the stats don't make a huge different unless you really want to min/max.

The most important thing to look out for, then, is how many genes your monstie has and which order they're in. Your goal should be to create a monstie with as many powerful genes and bingos as you can fit into it, and of course this all depends on luck: how many genes you get, which ones you get, what position they're in, and how they can be combined with genes from other monsties you have.