Verified answer

You have to be in a playlist that allows custom classes, such as infinity slayer (IS), big team IS, capture the flag, oddball, regicide, to name a few.

Playlist, such as swat, will not allow custom classes in order to balance that particular playlist.

If you're talking about not being able to choose classes in custom games, then there is an option that allows you to enable/disable custom classes within the game type editor, I believe it is under "base player traits"

Yes, in custom game Firefight and regular Custom Games you can make your own classes. You can choose what weapons and armor abilities you spawn with.