Hi Viconk,
I too am incapacitated, mine is due to extreme degenerative joint disease which does allow for some very short walks, but do understand your limitations better than most. I too love to take photos (can see my albums here if you want to) and do have an understanding - however slight LOL - of cameras, lenses, f-stops, etc.
From what you have said, two things you are going to need is a zoom lens (or very long prime, if you go the single lens reflex digital route), and a TRIPOD to be able to take slow exposures which allow more light into your lens.
There are a LOT of VERY good p and s (point and shoot) digital cameras out there, and they are probably your best bet with your budget, but make sure you have at least a 8X zoom. 10X or more would be better, obviously, but NOT digital zoom, optical only. Digital zoom you should not use as for the most part, you get that effect better on your editing software.
If you wanted to go dslr (digital single lens reflex), with that amount, the ONLY way is to go used, and yes ebay will have some good cameras with lenses for about that, but then you get into lenses and other accessories that you will want/need later, so that is up to you.
I will be glad to converse more with you, if you want, please feel free to contact me via this site, will be glad to "tallk" to you either by messenger or email!
Jacque