Realizing everyone has their own preferred process here are my 2 cents:
- with a cooler mash tun, opening and stirring every 15 min is just gonna cause temp loss with no real gain IMO. I vote preheat the tun well, stir the crap out of it the first time getting everything uniform, then leave it alone
- also IMO a mash out with batch sparging is unnecessary. In the 10 minutes you're doing your mash out you could have already had the first runnings heating in the kettle.
- it's a good idea to either measure the first runnings directly, or calculate the first runnings volume accurately (using known values for grain absorption and your tun deadspace) so you can batch sparge with exactly the amount you need. Dumping in extra water and leaving it behind when batch sparging is going to cost you efficiency points.
- you don't need to let it sit for 15 minutes with the batch sparge. You're just rinsing off the sugars, stir the bejesus out of it then vorlauf if you want and drain away.
- make sure to really mix the first and second runnings well before taking your preboil gravity and adjusting anything. It's easy to get a false reading from not getting the denser first wort mixed in with the second runnings.
have fun and good luck with it!