I think Moriarty would have a pretty low limit. He doesn't want people getting anything over on him. Low payouts, shuffle often, and he can go for blackjack (no hold on 17).
The Muddy Rudder seems to me like they might be playing more for the fun of it. Nothing outrageous, but considerably higher than Moriarty, just maybe harder to win? More decks or something like that. Really high bets, etc.
Underworld has that kind of lawless feel to it (well it does to me), so anything goes type gambling. High payout, low deck count. These are games between folks, played for everything or nothing.
For Canterbury Commons, they seem like the gambling kind, what with those caravans risking the deadly CW to trade their wares. Or maybe they use the games to fund their caravans. So maybe high bets, low-ish payouts, but relatively easy games.
Tenpenny strikes me as a roulette kind of guy, not sure why.
As far as integrating, those slots in Moriarty's don't look good there. They seem, well, tacked on. That little alcove where Burke is sitting would be a more likely place to have a couple slots. That being said, I also don't think he'd have more than 2 or 3 machines, them being hard to come by in the DC area. Really I think if any place is going to have any "official" gambling equipment, it would be Tenpenny.