Clauses
An Independent Clause (IC) is the same thing as a complete sentence: it has a subject and a verb. If you can put a period at the end, it's an Independent Clause.
Won't work
The FANBOYS are For, And, Nor, But, Or, Yet, So. Here's a big clue, though: The SAT basically only ever uses "and" and "but." Remember those and you're fine.
Not IC, then IC: comma.
IC, then not IC: nothing, unless the not-IC starts with which, a/an, an -ing word, such as, or among them.
Can only follow an IC. What comes after doesn't have to be an IC. It could be a list or explanation.
These words belong to the clause that's pushing back.
If ‘however’ or ‘though’ interrupts one independent clause (rather than sitting between two of them), surround it with commas.