Q:

Do I use one comma or two?

"I can tell who Hank and Bill are, but the fourth is missing, and the third's appearance is ambiguous."
Do I use the second comma or no? "The third's appearance is ambiguous," can stand on its own, but the "but" applies both to the second and third clause.

A:

You would put the comma before "and the third's appearance is ambiguous." You would put it there because it follows the rule of putting a comma before a coordinating conjunction that starts a new independent clause. 
The list of coordinating conjunctions would be FANBOYS: for, and, but, or, yet, and so. You know to put a comma if there is a subject in the second part of the sentence. The sentence you said, "and the third's appearance is ambiguous," would have a subject, and it would be "the third's appearance" since it is the thing that is doing or being something.
You would not have a comma if there is no subject, however, like in this sentence,
"I can tell who Bill and Frank are, but the fourth is missing and is ambiguous." (something like this)
But yes, overall, you would include the comma because you always put a comma when joining two independent clauses with a coordinating conjunction. If the sentence is very short, you may be able to omit the comma, like in this sentence, "He ate the pasta and she drank the wine." But it would still be perfectly acceptable to include the comma in this sentence, and if you were trying to write formally, it's recommended that you still use the comma.

