Not at the same time, no. What Japican and I did was send each other the vmf file through google drive. An alternative to that is having one person open the map, then his friends using teamveiwer to control some aspects of his computer. How much lag you had would depend on your connection type.

Spawn points: select your entity tool, then go down until you see "info_player_start", then place it on your map.
The button doors I think would be something like regular doors. Just make it so that they both have an instant move time, and if you press one, it does down and the other one comes up.
Converting to .bsp is easy, but might taks some time. It depends on your hardware like your processor and your video card. Alternatively, you can render with the "fast" settings, but it is prone to ingame errors and is not recommended for a finished map.