Is suitcase allowed in Metro and trains?

Yes, a suitcase is allowed on Paris Metro and Paris RER trains, but beware that many Metro stations have stairs you will need to navigate without the aid of escalators.  And some Metro entrances only have turnstiles rather than fare gates which may make passage with a large suitcase challenging.  Some of the larger stations have luggage pass-through slots, but this is rare except from the major Metro stations like in Chatelet-les-Halles area or Gare du Nord.