Why not go on Petfinder and look for a rescued rabbit? I bet you could find plenty of small rabbits up for adoption. They will already be spayed or neutered and health checked. Plus there are so many that need a loving home.
If you are going to put collars on them to stay on permanently, then you want the collars that can break away. The cheap dollar store dog collars would work well. I buy the small size ones for my Nigerians but since Pygmies would have a thicker neck, you may want to buy the medium size.
It is hard to tell in the picture. Are there any open wounds? It almost looks like he may have something stuck in his pad but it is hard to tell for sure. I hope he does alright for you.
I don't know if they do this for pigs but usually you want to do 2 Ivomec injections 14 days apart for mites.
Also, there is a "cream" called Nu-Stock. If you google Nu-Stock it should come up. Works great on all kinds of skin problems.
Have you ever copper bolus'd Koda? He could have a copper deficiency. Even with a good mineral, they can still be copper deficient. How much copper is in your mineral? I use Sweetlix Meatmaker goat mineral and it has 1750 PPM in it.
Also, mycoplasma could be your problem. You would want to use...
As far as disease resistant, you would want to find a good breeder. There isn't a particular dairy breed that is in general "disease resistant." Sanaans are considered the best milkers. But if you got Nubians, you could breed them to your boer male and the offspring are fairly desirable. Nubians...
With Safeguard, you want to worm him 3 days in a row and then again in 2 weeks.
Also, the block you bought is more of a treat. It won't have enough copper in there. You will want to find a good loose mineral specifically made for goats. My mineral has 1750 ppm of copper so you want to look for...
Safeguard (liquid) should be given at 1cc per 10 pounds. He looks small enough that you could bring out your bathroom scale, hold him and get a weight with both of you. Then weigh yourself and subtract. Also as far as Ivomec, you want the cattle 1% injectible and you can give it orally.
I would definitely have a large animal vet do a fecal for you and make sure they also check for coccidia. Even though generally adult goats don't have it, that is no guarantee. DiMethox is a better drug for coccidia. As far as Ivomec, the cattle injectible given orally is better. Give it at 1cc...
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I think stores in general are stocking less things to keep costs...
I have 4 barn cats and they don't bother the chickens. They would kill chicks but not the full grown chickens. You may want to consider getting 2 barn cats and not just one. They will be happier to have a buddy and can gang up on larger rodents.