Do check rides completed by part 141 schools with self-examining authority count like normal check rides for CFI renewal and gold seal?
No, check rides completed by a part 141 flight school with self-examining authority are not considered to be FAA checkrides for any purposes.

They do not count toward CFI renewal credit for maintaining an 80% 1st-time pass rate, and they do not count toward the requirement to obtain a gold seal CFI certificate.

My source is a direct answer from the chief instructor of a large part 141 flight school who received this information from the local FSDO.