So it doesn't matter if the locations are on different breakers as long as they run to the same fuse box on the same meter?

TRUE! I have 2 different breaker boxes in 2 different buildings, 200' apart, but on the same meter. It works great and is very clear.

Will the lock button stay on these intercoms without automatically shutting off?

Yes. You set one unit in LOCK and the other unit automatically receives if they are both on the same channel. Ideal for when you are using the units as monitors rather than intercoms.

Will this system work between main floor and basement?

It works through out the whole house as long as they are on the same electric wire. Please refer to Bullet point # 1 to 4 in the Product Knowledge... 2nd picture in the picture gallery for detail. The setup steps at the lower half of page 9 (SOLUTION: Electric wire) will also help you to find the outlets on the same electric wire

Is this for only two rooms or three rooms?

The 1,2,3 are different channels not rooms. You could have one in the living room, bedroom and bathroom. You could have the bedroom on 2 and the bathroom on 3. The living room can select which one he wants to talk to without bothering the other one. He could stay on 1 to receive calls from the others, they just select 1.