Excuse me, Hilbert's Hotel is infinite. That's it's definition. You want to talk about a finite hotel, you make up TemperedAlchemist's hotel or something, and what you say will be true of that.

That sum you mention is a sum of finitely many numbers. It's numbers from 1 to n, and n is expressed as finite. Therefore, it follows the normal laws of arithmetic, and shows that you can't do this trick at the downtown Hilton without pushing the penthouse occupant off the roof.

Similarly, you have room n and room 2n, where n is a finite number. If the room numbers end at n, it's not Hilbert's Hotel.

The rule is that, for any positive integer you can name, there's a room with that number, and every room has a different integer number. Any given guest is in a room with a finite number on it (or is near one in the hall). There is no last room. You seem to think that the rooms run from 1 to aleph-null, but that's not how it works. You seem to be thinking of aleph-null as a literal value.

For any positive integer n, both n + 1 and 2n are integers. (It follows that n + 2, 2n+1, 2n+2, and 4n are integers, of course.) Therefore, it's possible to move every guest from room n to room n+1, since room n+1 exists, and that leaves room 1 empty. It's also possible to move every guest from room n to room 2n, leaving the odd-numbered rooms empty.

It's true that one kind of infinity doesn't necessarily equal another kind (the number of integers and the number of real numbers are different, for example), but it's also true that no infinity equals something finite, which you were trying to do.