I have a 7 hour transit in DOH coming up (arrive 11pm, depart 6am). Is it worth leaving the airport at all? I was thinking of maybe going landside just to get a passport stamp, then going back through security and sleeping overnight airside. Any other ideas?

Not really, but few experiences could be worse than hanging around in that airport overnight.

There's not much to do in Doha at any hour, especially at that hour. I would recommend taking a taxi to Souq Waqif (which will cost less than USD 10) and sitting at one of the Arabic restaurants/cafes. None of them have great food or service -- and a couple are open 24 hours -- but it's a nice environment, and at midnight when you arrive, there will still be some activity. Then take a taxi back to the airport when you're ready, though taxis can be hard to find so allow plenty of time.

The bars in Doha are open until 2:00. There are decent bar/lounge options at a few hotels near the airport, especially the Sharq Village or Oryx Rotana.

Visited both the Business and First lounges last year. Food was OK, but I thought the wine selection was excellent. There was also a video games room, with no kids, which could provide some entertainment!

Transfering QR to QR, or interline to other carriers. Will your checked (if you have any) go across/through or do you have to go an collect it ?
We need to understand a couple of other things here. 1st/Biz or Econ = Premimum Terminal or normal terminal.
PT has all the attractions you will need to entertain yourself - food, drinks and some quiet couches to nap.
Normal Terminal, you may well look for a excursion into Doha.

Yes, transiting QR to QR, in economy with no status and hence no lounge access. My luggage will be checked right through.