Just play the lottery and it will up the pet's slvl upon doing so

As for finding eggs more easily, I can say the most effective method is probably just to join the discord if you have one. (You'll need to be verified to see the channels but Takkju or I can take care of that.) There's a userpage share channel where people will usually say when they need eggs hatched. It's slightly more effective but it works.

@Dreamcrystal