NFHS Rules
 

Does the run count if a preceding runner is declared the fourth out for missing a base, and the defensive team selects that out to it's advantage?

No. I don't have my rules book with me, so I cannot cite the rule number for you, but the whole purpose of a 4th out appeal is to nullify runs.

I think it is 9-1-1-f

Cecil has the correct reference. See also Case Play 9.1.1 Situation R.

I think it would be important to know how the third out came about.