Yes they'll stamp it in your passport. If you don't have one your visa will be cancelled when you leave Thailand.

You'd be getting a re-entry stamp. You can get it at your immigration office or the airport. I always get it at my immigration office incase of any issues at the airport where there is no turning back.

Yes I'd definitely say you are going to the Ministry Of Labor for your Work Permit. If everything is in order then you shouldn't be there too long. Nonthaburi Immigration isn't too bad either but get there early if actually do have to go there. I do my 90 day report there. It's a small office and can become very overcrowded and you could end up waiting all day for something that should only take a few minutes due to the queues.

That could be part of the problem. Some airlines won't let you fly with a one way.

Well what reason did the airport give for stopping him? Did he have a return ticket or a one way?

The only experience I have of this was a guy in the same queue as me in Vientiane back in 2012. He applied for an Education Visa the first day and collected it on the second day.

Yes it is fine. I did it last year. I had to go home for 2 weeks in the middle of my consideration period and was able to re-enter without an issue.