Can you say “Quid of the situation?” to ask “What about the situation?”

Coming from a midwest American English dialect, I can honestly say if someone asked me that question, I'd respond "huh?". It makes very little sense to me, even if it is acceptable grammar.

Better phrases would be:

• What does your father do (for a living / for work)?
• What line of work is your father in?

If I had to take it literally, I suppose I would interpret to mean any one of the following things (in order of confidence):

• Ethnicity / Race
• What my father is like (as in, good-natured vs. horrible, etc.)
• What my father does (work)

No.

Much is used more in negative than in affirmative contexts. A lot has no such restriction, and that's one reason why it's so common.

• I've never liked it a lot.
• I've never liked it much.
• I've always liked it a lot.
• *I've always liked it much.

Another reason a lot is common is because it doesn't distinguish mass from count nouns; there's a similar pattern for many, with plural count nouns.

• I never eat a lot of them at once.
• I never eat many of them at once.
• I always eat a lot of them at once.
• *I always eat many of them at once.

Much and many are on the NPI list, but they're not quite NPIs; there are a lot of affirmative environments where they can appear grammatically. Of course, as quantifiers, they already participate in a slew of idioms and odd constructions, so that's not really news.