I have an enemy I am trying to defeat in Explore that is kicking my pets butt every time. Is there any way to heal him or improve his stats while he is exploring? I finally got to level 4 so I really don't want to leave but my poor guy has almost no energy left.

No way to improve the stats, but energy does regenerate normally even on islands. So if you leave the pet there for a few hours, you can keep trying...