Anything before 1923 is automatically in the public domain. Some things post-1923 are, but it’s a toss-up.
As far as what you can and can’t do with things like cartoons that have well-known characters that are still under copyright in other mediums (popeye for example), this has always been a grey area. It seems like you can do whatever you want with it as long as nobody raises a fuss and it would probably take a lot of money and legal fees (although IANAL) to sort out who was in the right in that case.
I’m dealing with something similar right now in my profession… when the former USSR signed the Berne Convention (the international copyright act), it was not retroactive. Because of this, everything soviet that is pre-1973 is considered to be PD. The problem is that no one is quite sure whether or not that’s true anymore especially when applied to former soviet states which have now applied for retroactive copyright. It still hasn’t stopped me from trying to make a profit off of old Soviet classical music.