I agree with Kizza you definitely need to get him a proper cage and also not pick rabbits up from their back. When lifting your rabbit you should use two hands, one supporting the chest and one supporting the bottom. If your rabbit is small enough you can position the hand supporting the chest with you thumb over their shoulder for a firmer grip. Hold the rabbit with its head slightly higher than its bottom and with the bottom slightly tucked in. This will help prevent the rabbit kicking out backwards or trying to do a forward summersault.
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This is not picking up the rabbit by the back. This is picking up the rabbit at the end of the back, you are supporting the hips/pelvis area. I do this with my young rabbits as it is easier to transport them from spot-to-spot. However as the rabbit grows older you just want to take some precautions and make sure your rabbit it not to heavy to pick it up one-handed. Picking up a rabbit this way does not hurt them. However as an inexperienced owner I recommend learning different ways of handling, to see which way suits your rabbit the best. I do not recommend this way of picking up to a rabbit older than 6 months as thats when their mass starts coming in and that is about the time they will start needing more support. At this stage it should not hurt the rabbit as long as you are not holding the rabbit in that position for an extended period of time.
This is a lionhead. He looks young which would explain why his mane only looks like a "tuff" As he grows older he will grow out a longer mane around his head and around his body.Once his adult coat comes in the fur on his back will then be the desired normal fur. Some lionheads just seem fuzzy on their back when they are young, its just a trait they have to grow out of.
Here is an example of what I am talking about
This is Eevee when she was a young lionhead. Note how she has more of a wooly back and how her mane is still very short and not very defined.
This is Eevee at an older age, look at how much her mane came in and how the fur on her back is much shorter than the mane.
I would definitely say that this rabbit is a lionhead, seems like a blue vm. ( I think)
I would try getting your rabbit out in a cage as soon as possible, my 9 week old holland lop can already jump 3 feet (he is a very determined rabbit)
Never put a rabbit with a chicken the risk of coccidiosis sky-rockets. Chickens natrually have coccidiosis in their intestines so it could easily be transferred to your rabbit through the feces of the chickens. Bad idea, I do not recommend that. If that is your only coop trying separating an area(A small corner if possible) just for your rabbit, make sure your chickens can't get in and the small coop is very clean an disinfected. Also another thing chickens will do if put together with a rabbit is they will go after the rabbit. I had a 4Her would put a rabbit and a chicken in a coop together and the next day we got a call that the chicken pecked the rabbits nose off.
Hope I've helped, if you have anymore questions feel free to ask