What if an Executive of a Company whom you are in talks regarding your Job, stop replying your emails. What should you do?

No matter what happens, you have to be polite always. Send a polite email inquiring about your candidature. If you don't get a reply, send a polite reminder asking whether you have been selected/not favourably considered. If you still don't a reply, assume you are not getting the job and the Executive you have been communicating with is rather shy of saying an open no. Write a polite thank you letter and close the chapter...