Can I leave Thailand briefly to apply for a DTV visa and then return?

Think of it this way, in the old days you would go to the Embassy and hand in your passport and wait for a sticker Visa. It would be impossible to travel outside that country when not in possession of your passport.

The DTV is a tourist visa. You have to remain in the country you apply in until it's been granted. I applied for a tourist visa in the UK using flight details of a flight I wasn't booked on as I wanted the visa before I booked flights. My company then contacted me and asked me to fly to Libya for a few weeks, I agreed as long as they flew me straight to Thailand instead of the UK after I'd finished the job. While I was in Libya they asked for flight confirmations so thinking nothing was wrong I sent them my flight details from Tripoli to Bangkok. My visa was cancelled for leaving the country of application.

The quick answer to your quick question is: No.

Understand that if the embassy you apply at gets any notion you are not in the country they will ask you for a copy of every page in your passport, including all the blank pages.

Why can't you leave Thailand for more than 4 or 5 days?

The answers you get here will be exactly the same as the answers you got on the Destination Thailand Visa (DTV) group. Remain where you apply until the process completes