Is it normal for my Mac to get hot while playing games?

Most Macs has a really bad thermal design. It's normal for it to get hot by doing anything more intensive than browsing Facebook. If they get too hot they will thermal throttle, which means they will lower the performance to prevent damage from overheating.

This is how it looks like inside my own Macbook Air 11". You can see the tiny little fan at the bottom left corner. It blows the hot air out the back of the computer, and it pulls cold air in from.... wherever it finds it's way in. I can't really find any reasonably engineered air intake in this computer. I decided to put on Noctua NT-H1 which is one of the best thermal pastes out there.

Also, the green stuff is conformal coating that I did to prevent water damage. This computer was water damaged when I bought it, which is why I got it cheap. I have fixed it and it's now fully working. I don't recommend opening up your Mac if it's under warranty.

What you may want to consider is a laptop chiller, which is basically a bunch of fans blowing at the underside of the computer. It may help a little bit.

Yes, they will get hot playing games. The battery will drain quickly if not connected to the AC charger. All CPUs in Apple computers will shut down if they reach thermal cutoff.