There's no Thai embassy in Thailand as this is (already) Thailand, just like there's no embassy of France in France. Embassies exist only in foreign countries. You can get a 30 day extension at immigration offices in bangkok or a 60 day single entry tourist visa at a Thai consulate in a neighboring country such as Laos.

You're not clear in your comment: there are 2 spots where your father can be denied to proceed: A. The airline desk or B Customs. If it was because of no return ticket, he would know it because airline staff would tell him. Besides, it's extremely rare that airlines make a fuss about this. It's not a visa issue since A, he doesn't need one as he can get into Thailand on the 30 day exemption and B, he has a SETV. In my opinion, it has to do with his past overstay or something happened at the airport, like him maybe having a few drinks before the flight. Note that he could also be blacklisted by that particular airline for an on-flight incident in the past. If customs stopped him, it would have to do with a serious legal concern in the UK in regards to your dad, like being on probation.

Not true... It depends on the airline