As for the breeds, many cats are DSH, DLH or mixed breeds, so it's hard for them to determine whether the cat would be prone to common genetic/breed ailments.
Maybe you'd like to ask them to review their adoption contract before visiting with the kitties, many of the things you are requesting ARE in the adoption contract. I have 14 days to return Fenway if something should happen...I can return her for a refund of my adoption fee if something should happen and she becomes ill.
Legit rescues should provide you medical history, even if they don't have anything prior to the animal arriving at the rescue/shelter, they should do testing for FIV/FeLV and up to date on vaccinations. Spay and Neuter should be done, but if the animal is too small or too young most will ask for a deposit as an incentive for you to actually get the s/n done. That's what happened in my case with Fenway. Which for me, is fine because I will absolutely take care of that, but it still worries me that they do that because I know there are a lot of irresponsible pet owners out there...
There are a lot of members here, so there may be someone who knows of a respected rescue or shelter in your area, if they do, I'm sure they will reply soon.
Takin ga trip to your local humane society is not a bad idea at all!
As for my baby that didn't make it, I absolutely agree with you that it's horrible that she had to go through what she did. Honestly, as much as I fell in love with her and wanted so badly to save her, she probably should have been euthanized at the shelter, as she just wasn't well and she was badly emaciated (scale of 1-5 with 1 being the worst, both vets listed her as a 1), had watery diarrhea, horrible ear mites and wouldn't eat without being force fed. Though, it does make me feel better to atleast know she ended up with someone like me who knew what to do and had the means to give her the best care and keep her comfortable rather than someone who couldn't do those things...So, at least if she was going to be adopted to *someone* I'm glad she ended up with us.