Sorry to hear about your medical issues and hope you get better with good medical treatment.

You mentioned a lightweight rifle with low recoil for hunting Coues deer, javelina and rabbits. I checked Arizona's laws and it looks like they only require a centerfire caliber for hunting big game so it looks like you're on the right track, caliber wise. Some of the guys have already given you pretty good suggestions on 22 calibers and some of the barrel length requirements. Personally, I think you'd be fine with a .223 as it's plenty of caliber out past your self imposed limit of 200 yards and you could find lots of fairly inexpensive FMJ ammo for target shooting/plinking.

What weight are you looking for? You could probably find a .22 centerfire rifle that weighs 5.5-6.5 pounds fairly easily as there are mountain rifles made and others made for smaller people. (Edit--Kimber does make a rifle under 5 lbs but it's pricey at around $2K. Savage makes a lightweight hunter model at 5.5 pounds but it's listed for almost $1K).

I 2nd the advice on a Limbsaver pad. I put a slip on Limbsaver pad on a 45-70 and it makes a huge difference. I was looking to add a little length of pull because of the short stock and my buddy and I were very impressed with the felt recoil reduction. As a bonus, you can simply slip it off whenever you like and it's fairly inexpensive to purchase.

Something else you may want to consider is buying a rimfire rifle for hunting rabbits. Ammo is cheaper and they generally tend to weigh less than centerfire rifles. I know it's buying 2 rifles but hey, you did say you were getting an itch to go out hunting so why limit yourself to just one gun. Another possibility for the rabbit hunting is a rimfire pistol, scoped or not to your taste. You could wear it in a holster and not have to "carry" it on a sling.

Hopefully this helps give you some ideas to consider. Good luck with your hunting and let us know how it works out.