Did you get the new gecko from the same place as the old? Then I would guess they may have the same thing, either illness or parasites. If the vet couldn't find anything (which happened to me) find another vet. Call a herp store and ask where they take there animals. Also, realize if you didn't sanitize the enclosure between animals, you could have infected the second from the first.
If you say you are using the lamp for the day time and the UTH for the night time, sounds find for a 10 gallon. The UTH should be 1/4 of the area on the bottom, the packages are good at telling you what size to get. so I'm guessing that is right. Temps sound fine.
Soaking in water for dehydration? Never heard of it... seriously, they are desert animals, they aren't going to soak up any water through their skin. Now, if they are having a bad shed and have loose skin attached, some people place them on warmed and soaked papertowels to help the process along.
Geckos don't throw up, some have said they can from too much stress, personally I haven't seen this. I have however seen geckos throw up from illness and more likely parasites. Simple put, the parasites block the system and the gecko can't get food down. This can also happen with sand impaction, are you using a proper substrate for him like paper towels or tiles? Young geckos fill their systems with sand and back everything up that way. Pet stores are notorious for selling calcium sand and saying it is perfectly safe for young ones, it isn't and it is sad.
OK, to your specific questions and off my soapbox. It is not normal for geckos to skip meals until they get older and stop growing. Maybe an occasional missed cricket, but definitely not throwing one up. However, since geckos do eat their skin after a shed you may notice a reduction in appetite or a skipped meal if he shed during the day or in the evening. (Are you sure?) I'd say appetite for our girl returns in about 12 hours after a shed when she was a wee one. So, if she shed early enough, she'd eat the same night... but she was also a little piggy too, your milage may vary. 
Gecko shed rates are based on age because of growth rate. Young ones are usually in the range of 2 weeks or sometimes less and adults around a month or so give or take a week.
I haven't heard the sunburn thought before. I have handled my gecko immediately after shed to inspect toes and such and she has never given any indication of pain or not wanting to be messed with.
Good to hear he is drinking, that is a good sign at least.
What I am worried about is that you said he is walking tenderly... Now, just woken up geckos have that "just woken up where is my coffee" walk sometimes, but are you meaning his walk now is always slow and careful and wobbly. I ask this because this can be an early sign of another scary thing, Metabolic Bone Disease. It is a lack of calcium. At this point it wouldn't be at all your fault, you haven't had him long enough, it would be a lack of care from who you got it from. Are you currently gut loading and dusting your crickets everytime with a supplement like Leopard Gecko Cricket Dust by TRex?
I don't want to scare you, but your gecko is exhibiting some strange signs. I hate to tell you another vet bill is in order, but it would be good to have him tested for MBD and parasites. Like I said, find a reptile store and ask where they go, that is how we found good vets that knew what to do. All of what I mentioned can be totally treated and fixed if caught early.
I wish you the best of luck. I'm pulling for the little guy, let's hope it is just stress or something I didn't think of because I know my message is a real downer. I apologize, but I desperately want you to know the worst of what it could be so action is taken as quickly as possible. Good luck, please ask any more questions you have, we are here to help and I will try not to be so doom and gloom next time. But my story is similar to yours, first gecko wouldn't eat, we sucked down a $100 vet bill took him back to the store, got another, had a wonderful 9 months with him until we moved 8 miles and the stress upset the balance between him and his parasites. He had been living completely symptom free for 9 wonderful months. Better vet, hand feeding, but eventually we had him put to sleep, I will do anything to keep anyone else from going through that.

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0.1 Albino Leo Gecko
0.1 Crested Gecko
1.0 Dear Boyfriend
Departed: Harvey and Spock