The link shows a variety of BBs so one can't really say if any of them is right for you. I would say that if your current BB is correct for your frame and intended chain line then you are safe in buying one with the same specs. Likewise if your BB fits the other frame - and the axle length is suitable for a proper chain line. Your other frame may have a different shell width and thread so you would need to check.

Also, yes one side will come off. You would not be able to install it if it didn't.

Here's a link to some info that may be helpful: Sheldon Brown's Bicycle Glossary Bo--Bz