Is there any difference between saying ‘mae gen i gar’ and 'mae gar ‘da fi’ ? Also, with the former have I sometimes heard people drop the mae and just say ‘gen i gar’ ?
No, they’re exactly the same meaning. It’s just that you’re more likely to hear the “gen i” construction in the North and the “'da fi” construction in the South.
And yes, leaving off the mae before gen i is pretty common.