Questions: Is the gumming and swallowing ok?
Yes...sounds like it's just fine as long as whatever you give him is soft enough and small enough that his digestive system won't have a fit trying to work harder to do the job that his chewing would've.
Could the baby teeth still get pushed out?
Sure! Realistically though? No. Those disiduous teeth most likely aren't going anywhere and will need to be pulled.
How long does the teething process take?
Varies GREATLY from dog to dog. I'd say he's through that process.
The reason (if I had to guess) that he's in pain is due to the fact that a few of those disiduous teeth have been pushed to the point that they may be pointing towards his cheeks and when he chews on anything, guess what? The inside of his cheeks are getting scraped. Take a look and check for yourself. It's quite common w/ that many retained! Can they develop behavior problems due to teething?
?? I don't understand what you mean? If anything, not allowing him to chew on things that you give him would only encourage him to take out his need on something else. And I'm quite certain you two won't agree on what he decides to chew on if that choice was left up to him!