Yes, they can't put you in a VIP room and don't bring champagne, but treats in the form of bonuses and special offers are even better if you ask me.
They'd probably receive the same rewards (or maybe better) since there are different ways on how rewards can be calculated and afaik house edge is higher on physical casinos.
Maybe, if casino owners knew. My point is that as far as I've always seen it, in physical casinos, you are considered are "VIP customer" just when the management and staff already recognize you (same as if you come to a certain restaurant often, you can get discounts, because the owners already know you), but this can be tracked way better in online casinos. There are certain rules: e.g. make 10 bets worth 10$ each and get X free spins. After a year of being registered - you also get certain bonuses, same goes with holidays, your b-day. I just can't see how this could work with offline casinos. The max you can get is maybe some sort of client's card, so they can track how many times have you visited them, but other ways of recognizing client are complicated.
It is true that in online casinos everything is automatic and the system will take care of the bonuses, and as you say if there is a card for frequent clients at physical casinos this could be a way for them to acomplish something similar, however for what I know this is done by the managers, basically there are people that their only job is to watch the players and see how big your average bet is, how long have you been there and if you are winning, and if they think you are a client worth giving a bonus they will do so in order to try to gain your preference.