No, but most modern proof coins do. There are a few errors and a few special coins that don’t, but for the most part, starting in 1968, proof coins has S mint marks.

No. Since 1968 MOST proof coins have had an S mintmark but there have been exceptions. The proof 1999 SBA dollar was a P mint, proof silver eagles have has S, W and no mintmark proofs. Depending on the issue proof modern commemoratives have come with P, D,S, and W mintmarks. Before 1968 Philadelphia made all proof coins (with a very few special exception pieces) and they had no mintmarks.

It's not a proof. It would have to be a rare no S error, and then broken out of a proof cent and spent. And then, you would have to find this one coin out of countless billions. It's probably just shiny. Proofs have better details and often mirrored surfaces, and squared rims. I am not aware of any No S 2014 proof cents.

Proof coins should 1) have a mirrorlike background field 2) have a VERY good strike with clear devices and 3) have the edges if the coin squared off. This of course would not hold for matte proofs or reverse proofs.