it can take 5 or 10 days, and you need to remain there until the process completes. Why not an ED visa if your son goes to school? Have you checked with the school if they will be OK with him on a DTV?
You can have someone go on your behalf, you don't have to go in person. Just make sure you have all the relevant documents to give them. If you are early enough you can do it by post and if it's not your first filing since entering the country you can do online
Unless you want to apply for a 12 month extension when you return for your 2nd trip I wouldn't get a re-entry permit, get another 90 day Non-O while in the states. If you do want to apply for a 12 month extension make sure you put funds in a Thai bank on the 1st trip