First of all, night temperature isn't that bad - easily gets into the minus numbers in the wild and they still manage to survive. Don't leave any heat on at night, it's not good for them, their body needs a bit of down time.
to get a thermostat, especially when the weather here gets a bit warmer you will have real risks of overheating him. Dimming thermostat is best for bulbs. You said he spends a long time basking, this indicates that he's not getting enough heat. Where were you taking that 38C and 31C from? Directly under the bulb at the highest point where the beardy sits to bask could go up a few degrees (40-45C). If the temperatures aren't high enough, he won't want to eat.
The eye thing could have numerous different causes, someone already said sand but others are that the crickets have been at his eyes or photo-kerato-conjunctivitis (caused by getting too close to the UVB). Looking at the way the tank is setup (mostly that corked bark) that could be a possibility. Not opening his eyes could well be the cause of why he's not eating - if he can't see, he won't hunt.
UVB wise I don't know why you have a 10% and 2%... knowing [email protected]
they probably think that it equals out to a 12% bulb. I would leave the 2%, they won't be doing anything for the beardy. If you're able to have a look into getting Arcadia T5's, by far the best UVB out there.
Take the sand out for now, you've already said that he had a mouthful and calci-sand is one of the worst substrates. Until you have him eating well I would put him onto something like kitchen roll, newspaper or lino. How does his stomach look, bloaty or lumpy?
I can't remember whether you mentioned any supplements?
They do suffer from relocation stress, some for a few weeks. Keep handling to a minimum and only go in the tank for maintenance/feeding for now. However if he appears dehydrated (sunken eyes, wrinkled skin) then it may be best to give him a bathe.
EDIT - in answer to the previous couple of posts:
It's not recommended to feed them very large food items - that in itself can cause problems, but if the heating isn't sufficient too than you may have a problem.
The white part of the poo is urates (their version of urine), nothing to do with calcium.
No underbelly heating should be provided, by the time they sense the heat they may have sustained serious burns. Better safe than sorry with that one.
They shed pretty much from day 1, shedding depends on age. However if he is darkening it could be because it's colder or due to stress.