Can I Get Sick From Dead Mice In The Walls?

Erm, yes, of course. The bugs will be trailing all kinds of lovely bacteria all over your place, and you will be picking up the germs on your hands, food and clothes. Do you have a public health authority in your country? If so you need to contact them to come and clear out the mice and bugs. Or your landlord. I don't know what your housing situation is, but you can't live like that, it's unhygienic and potentially dangerous for your health.

Oh, come on.  We live in the country and get a mouse or two in our attic every year.  They eat the poison, they die, and if you can't find them to dispose of them, then you live with the smell for a while until they mummify in an area you can't get to anyway.  They are a part of life in the country.  As long as they aren't trespassing into an area where you both cross paths.  C'est la vie.