"The other man is not rich, either."
Translation:A másik férfi sem gazdag.
The rule is that se is used in imperative sentences, and sem in all other cases, but in spoken language (with the exception of imperative sentences, where you can't use sem) they are interchangeable. So, you're first sentence is natural in spoken language, but try not to use it in writing.
Your second sentence is incorrect; se and nem can only stand together if they mean "neither ... nor". "A férfi se nem gazdag, se nem szép."
I am not clear on when másik should be used and when más should be used.