Does anyone have any tips on beating the monsters on the island? I know the combat is based on paper rock scissors style, but like, is there a cheat sheet for what a monster is most likely to do?

Hmm, we don't really have one of those, its more like trying to see what kind of move (blow, bite rush) effects them the most, like for some, rush is very effective, while for others it doesn't do a lot of damage. Thats more from my personal experience though, someone might have a cheat sheet, but its pretty random with what effects them the most and whatnot.

You can can choose a pet with a special gene/attack, for example Ice Breath will replace the normal bite attack and increase bite damage by 50%. In this case, you may want to use the bite attack more often. 

There is an answer under FAQ >> General Gameplay questions >> Any tips on winning the battles? that you may find interesting, but probably not very useful.

Finally, you can check under a pet's profile Stats >> Characteristics and look at their Bite/Blow/Rush damage. If a pet has one or more stats in green, you might want to use that attack more often. If you look at this pet, you can see that it's rush damage is signficantly greater, so I would use the rush attack more.