What's the difference between 番 and 番号?

Question

Could I use 番 instead of 番号 here?

私たちの部屋の番号は三十一です。

Answer 1

Could I use 番 instead of 番号 here?

I'm afraid not. 部屋の番 would mean something different than "room number". (It would be "watching/guarding a room". cf: [留守番]{るすばん} "house-sitting")

○ 私たちの部屋の番号は三十一です。(correct)
× 私たちの部屋の番は三十一です。(incorrect)

You can use 番 as a counter like this:

私たちの部屋は何番ですか？ Lit. What number is our room?
私たちの部屋の番号は何番ですか？ Lit. What number is our room number?"
(rather than 私たちの部屋の番号は何ですか？)
私たちの部屋は３１番です。 Lit. Our room is number 31.
私たちの部屋の番号は３１番です。 Lit. Our room number is number 31.

Answer 2

Not likely.
番 by itself is mostly for expressing order, like first second or third.
番号 is for explicit number like a room number.

However there are other ways to express what you have said.
If the subject building is known by both the listener and speaker then 31号室 can be used.
Also as an extension to the answer. Implied possessives do not always need の. In this case 部屋番号 would be fine.
Further, Japanese is quite subject indifferent so using 私達 is not even necessary.
For example, in responding to the question of what room you are in you can say "31号室".
If stating your room number then: 部屋番号は31です。

Answer 3

No, because 番 itself doesn't have a "number" sense. It literally means "duty, shift, turn", and the usage X番 "number X" is a spinoff from the "turn X" meaning. Thus you have to use 番号 when you generally refer to "number" without a specific number X.

There are indeed a handful of combined words that contain lone 番 to mean number (型番, 付番, 地番 etc.), but they were originally abbreviated terms and cannot be taken as the rationale that 番 has a general ability to mean "number".