Besides gun game? I was watching some YouTube on the game and noticed a death match mode with random weapons that looked like it had respawn after death enabled. But any other custom modes I should look out for in the server browser?

There are some servers that run straight team deathmatch mode, but it isn't an official mode you can filter by. Look in the browser for deathmatch in the server name. They have unlimited respawns and allow you to use whatever weapons you want. This is how many serious players practice various aspects of weapon handling, movement skills, and other physical aspects of the game.

Yeah the deathmatch servers are the way to hone your shooting skills. You can search by keyword on the browser so dm or deathmatch should work. In CS:S and 1.6 there were quite a bit of different kind of DM games: free for all, normal spawn and random spawn. I don't know if this is the case in GO.