Just got first cat from shelter, seems sick, HELP

Hey guys.

So I just got my first ever cat, and I picked him up from my local Humane Society shelter. Apparently he was a stray, he's about one year old, and they told me he'd only been there 6 days before I snatched him up.

When I was playing with him at the shelter, I noticed he was soooo chill. By that, I mean he's very friendly, I can play with his paws and everything, he's affectionate like crazy and has a beautiful coat. One thing I noticed was that he was always purring, which I thought just meant he was really mellow.

Now that he's been home for about 20 hours, he purrs non stop. In fact, I think it's a respiratory thing because he never stops purring...both inhaling and exhaling. He also coughs/sneezes (not sure which) once in a while. Not lots, maybe four or five little coughs an hour...if that.

Here's the part that worries me. He hasn't touched ANY food, I've dried dry food and wet food, and I've tried to heat up the wet food so that it's more smelly.

I took him into the bathroom and ran the hot shower for steam, in case he had a blocked nose and couldn't smell the food. That did not help.

At the shelter they told me that he had recieved a Rabies shot hours before I adopted him, so perhaps this lack of appetite is a result of that shot? With the coughing and stuff I worry he has URI or something. His eyes appear somewhat runny, but not really bad...and his nose is not goopy or runny at all.

He also has not used the kitty litter even once. I checked around the apartment for ninja turds or secret peeing spots, but nothing anywhere. My theory is that he hasn't eaten, hasn't peed, hasn't taken a crap. He did, however, empty one small dish of water during the night, so he's drinking.

I'm a little worried. He scratches his post, and likes to be petted, follows me around sometimes but also sleeps a lot. I know cats sleep a lot, but I can't decide if his behaviour is normal or if he's sick.

I'm a first time cat owner, so I'm being freakishly over protective and obsessed with every little thing he does, but I can't shake the feeling something is wrong with him. He's also super skinny, but as I said he was a stray street cat for one year and only lived in the shelter for less than a week...so he could be lean from living in the wild.

Help me, you guys!