Quote Quoting nogusielkt
1: Is not being allowed to drink always part of probation, or is that a case-by-case basis?
It's usually case-by-case (some states or counties may have a blanket policy; I can't speak for every U.S. jurisdiction), but is a very common term of probation.

Quote Quoting nogusielkt
1a: What kind of punishment would he get for drinking, and drinking with a minor (whatever the technical term is)
If he violates his probation, he could be resentenced on the original charge. If he is furnishing alcohol to a minor, he would likely face an additional criminal charge for that offense, as well as probation violation proceedings.

Quote Quoting nogusielkt
1b: What kind of punishment might happen to my mother (home owner) who would be at work during this time?
Probably nothing, unless she is somehow complicit in furnishing alcohol to a minor.

Quote Quoting nogusielkt
1c: Would his probation officer come to the house if called to check on this?
Most probation officers are content to pretty much stay in the office, but some do check up on their probationers. So it is possible.

Quote Quoting nogusielkt
2: Who generally gets punished if drugs (guessing marijuana) are found in the house during this process?
Whomever can be shown to be in actual or constructive possession of the drugs.

If all of this is really a concern - that the boyfriend is going to violate his probation, serve alcohol to minors, and bring drugs into your mother's house - perhaps she should consider whether this relationship is worth pursuing.