XBOX 360 in the Hotel
Solution 1:
Depending on the hotel set up (I used to work for a company that did this exact kind of thing), you may or may not need to have your Xbox passed through the connection so you don't get billed twice (unless it's already free).
There is no difference, but if you call the help desk, they may put through your xbox directly (if you can provide them the MAC address) we used to do it since they already paid for the connection, we just look up the IP assigned to the MAC and then put it through for the same amount of time as your room time is left.
Having the xbox connected directly to the connection means one less hop when it's communicating to the outside world. While this extra hop is usually not a big thing, it's one extra step you don't need if you can connect directly to the Xbox itself.
As far as the disconnect issue, it is more likely a network issue (i.e., not enough bandwidth, bad connection in the room, etc.).