That vet actually sounds like they might have been pretty decent. I'm glad you were able to get the bun looked at. Mites does sound right. I'm glad there doesn't appear to be anything else wrong. Mites can usually be pretty easily treated when the correct med is used. I'm guessing the vet gave an injection of ivermectin? You may want to double check the correct dosage was given as too much is bad and too little won't be effective. Rabbits need 0.4mg/kg. So the amount given will depend on the concentration of the medication. A follow up injection in two weeks is good, but I would suggest also getting one more two weeks later. It helps ensure the life cycle of the mite is completely interrupted. Sometimes even a 4th or 5th dose is needed if the infestation is very severe, so something to keep in mind if you aren't seeing as good a recovery as you would like. Also sometimes the injections may need to be stepped up to every 10 days, if you see the mites getting better but then getting worse again by day 14, so just something to watch out for.
http://www.medirabbit.com/EN/Skin_diseases/Parasitic/Mange/Sarcoptes.htm
One thing I will
strongly suggest NOT doing is not bathing the rabbit. Bathing a rabbit can actually be quite dangerous. For some rabbits it can actually put them into shock and could prove fatal. It's just generally seen as a very bad idea for rabbits, in most cases, and is only considered in a very
extreme case, and even then must be done very carefully with careful monitoring of the rabbits stress and body temp levels. The ivermectin injections should actually take care of the mite problem completely if done at the right frequency and for the appropriate amount of time. And you seem on a good schedule with that if you increase it to 3 doses 2 weeks apart.
With the cream, does it say what is in it? Just want to make sure it is rabbit safe, as rabbits will groom and ingest anything put on their skin and fur. Also it might be good to find out what the purple mite treatment on the skin contains in its ingredients if at all possible, as you might need to wipe off with a damp towel and dry as much as possible. Though I'm sure at this point that the rabbit has thoroughly groomed and ingested whatever it is, so hopefully nothing in it is toxic to rabbits.
It doesn't look terribly thin to me but does look a bit on the lean side. You can look along the back and kind of see the indent along the back muscles. That should all be filled in for a rabbit at a healthy weight, but that should fill in as the bun gets on a decent diet. You may want to look for a different rabbit food. Muesli mixes are actually not considered to be very healthy for rabbits, and can actually lead to increased digestive and dental problems. The best thing to look for is a plain hay based pelleted food. You probably won't be able to get timothy hay based pellets, but if you can find pellets with alfalfa hay as the first(or at least second) ingredient, those should be a decent enough pellet. You can always post the ingredients on here for an opinion. Typically pellet ingredients here in the US will have alfalfa, a grain ingredient, soy hulls and meal, molasses, and added vitamins. Usually it is best to gradually transition a rabbit onto new pellets, starting with a small amount and gradually increasing, but if you have no hay available yet, then the bun does need to be getting some food, and veggies can only go so far because of the high moisture content, so I suppose just do the best you can. Amount wise, around 1/4-1/2 cup pellets per 5 lbs(2.2kg) body weight is a good amount when the rabbit is also eating hay. You basically ensure the rabbit is eating a pile of hay the size of it's body, and adjust the pellet amount accordingly. So if it's not eating enough hay and/or it's fecal poop is still looking small and hard, then reduce pellets and make sure the bun is increasing it's hay consumption. If it is eating the pile of hay and is running out of food, then increase pellets and/or hay. A rabbit can basically have as much hay as it wants, and that is good for them, but pellets are just there for vitamins and weight gain. So if the rabbit is not putting on enough weight, you can increase pellets a little. It's a bit of a balancing act, finding the right balance of hay for good digestive health, and the right amount of pellets for good weight maintenance. Do the best you can. You kind of learn as you go. If you stick with the general guideline for body size pile of hay and 1/4-1/2 cup pellets per 2.2 kg, you should be fine.
With the nails, you can get some clippers and try to clip yourself. It can be hard if you haven't ever done it before, as rabbits don't usually like to cooperate and you have to be careful not to cut into the quick, as well as holding them in such a way that they won't freak and injure their back. Might be worth having the vet do it when you take the bun back in. Usually has to be done every 8 weeks. You could also have the vet sex your rabbit, and if the vet knows how to check rabbit teeth for sharp points, that may also be a good thing to have done as a general checkup.
With suddenly adding that much lettuce(a dark leafy lettuce, not iceberg) into the diet, you will want to keep a close eye out for mushy poop, but it sounds like the bun is adjusting well. Once you get the hay, that will help. Keep in mind though that rabbits eat a lot of hay. They should eat about a pile of hay the size of their body per day, so if you are getting a small bag of pet shop hay, that can be expensive, at least here it is. The bigger the bag, usually the cheaper it will be. The best thing is to buy horse hay if you can find a good quality grass hay there. Maybe a racing stable, and just buy a few flakes off them? Otherwise I guess you would have to stick with the pet shop hay, which is fine, just usually much more expensive
For the litter, you will want to make sure it is rabbit safe. Clumping and clay based cat litters are not safe, nor is cedar. Usually a pine or paper based litter is good. Pine shavings are only ok if kiln dried, but if that is all that's available then that's ok. You just have to keep in mind when using litter, that rabbits will chew on everything, so whatever you use has to be safe for them to nibble on. The best thing to do would be to try to litter train. Just get a rectangular cat litter box or plastic storage bin that will fit at one end of the cage, and put the litter in there. Then you can line the rest of the cage with newspaper. You can also use fabric that doesn't have loose threads that can possibly be ingested by the bun, but usually in warm weather, rabbits prefer to lay on a smooth cool surface, so the newspaper works well for that. The vet was right about keeping the bun indoors with those temps. I had absolutely no idea it was so hot there. Temps that high can actually be fatal to a rabbit. I'm glad the bun was ok and is indoors now. But even indoors can get hot for rabbits. Generally if temps are above 80-85f or 27-29c, then you need to take steps to help a rabbit cool down. An easy way is to offer a pop bottle filled with water and frozen. Then if the rabbit is hot, it can lay next to it to cool down.
I want to commend you. You've done a great thing helping this poor bun. I hope you are able to keep him/her(?). It looks like you are on the right track for this bun to be healthy in no time
Mites do make it look to be in a more sorry state than it probably is. Usually as long as a rabbit is eating well, it is not doing too badly. And it sounds like it is eating fine, but keep an eye on the poop. If the pooping stops and the rabbit then stops eating for more than 12 hours, you need to get the bun to the vet for a possible gut blockage. This is why hay is so important in a rabbits diet.
http://www.bio.miami.edu/hare/ileus.html
The cage looks big enough to use for now. If you do keep the bun, you can expand the area with an xpen if you want, or some people let there rabbits free roam, though in that case you have to rabbit proof, as rabbits tend to chew on things, including power cords. But depends on the rabbit. Some aren't big chewers. Anyways, good luck with the little guy and please keep us updated on how it's doing.