He DOES NOT need a heat lamp, let me tell you that much. I mean, if you got the extra money, but he in no way needs it!
To tell if he's sick, see if he's acting different - lethargic, odd poops, peeing on himself, ya know, the usual
Yes, check for fleas & lice & ticks.
I don't see much improvement - maybe try adding toys, though. Like cat balls (the plastic ones with a bell inside), wood (like parrot toys) Etc.
Carrots are sugary, that's the issue. Otherwise they're fine. But our rabbit likes celery, & broccoli a lot. A quick google search should give you a good list, here's one anyways
Grass (without any chemicals)
Celery (strings removed)
Alfalfa sprouts
Carrots and carrot tops
Herbs (basil, parsley, cilantro, mint)
Bok choy
Dandelion flowers and leaves (untreated)
Kale
Radish tops and sprouts
Broccoli (mostly stems and leaves)
Pears
Blueberries
Grapes (no seeds)
Raisins
Banana
Apple (no core and seeds)
Plums
Raspberries
Strawberries
foods to avoid :
Green beans
Potatoes
Beets
Avocado
Cabbage
Sweet potato
Corn
Onion
Rhubarb
Cereals
Crackers
Here's the website, it lists more
http://allaboutbunnies.homestead.com/Things-bunnies-can-and-can-t-eat.html
Avoid House Rabbit Society. Imagine a place where people own pet chickens, & think its awful for you to eat chicken, (but ok to ear any other type of meat, even if its factory raised and lived in horrible conditions), feed chicken to your pets or breed chicken. Same thing, only for rabbits. Yeah. Don't go there, & don't join the site. That's what I wish I knew!
Ill get some pics of my hutches, though I breed for meat so they might be considered small. Mine are played with all that much, but I'm going to move the, to a colony setting so They'll be happier.
Very pretty rabbit you've got there, btw. We have 2 Rex mixes, though only one has Rex fur. She's due to have her first litter soon, too, yay.
works if he's a Rex, or has Rex fur (I know many Rex mixes do). 
