How to care for found cat without harming indoor cat

Hello,
A cat has been living under the bushes in my yard for about a week. I could not get it to eat until I bought some Fancy Feast today. Now he's out of the bushes and acting like a pet - asking for belly rubs and sitting on the porch all day. I strongly suspect that it is an abandoned pet. It is too clean and healthy and friendly to be a stray.

I have notified all of the shelters in the area and everyone in my neighborhood. No one has claimed it.

It is young, but very mellow and affectionate. I would like to take it to the vet to confirm that it has been neutered. I probably won't even need a trap - a borrowed carrier will do.

My question is about keeping my own indoor cat safe in the meantime. How do I avoid transferring diseases? The cat looks very healthy, but looks can be deceiving.

I have been washing my hands after petting the outdoor cat and have been careful to avoid cross-contaminating food. Is there anything else I should be doing?

Does anyone have experience with introducing an abandoned cat to a very spoiled indoor cat? I have had him since he was weaned. I have read about cat introductions, but I'd rather not go down that road if it is unlikely to succeed.

Thank you for your advice.

One more question: should I leave a litter box outside for the cat? I haven't seen any mess, but I would rather proactively prevent it. I don't think this cat is going anywhere anytime soon. You would never guess that it doesn't belong to me if you saw how comfortable if I'd around me and how it doesn't stray more than ten feet from our house.

You should bring the cat to a vet and have it scanned for a microchip.  They can tell you if the cat has been spayed/neutered at that time and also test for diseases and give it a good flea treatment and possibly worm the cat.

Bring the cat in but keep it in a room away from your cat for at least two weeks.  If it has a URI or virus, it will show up during this time period.  You also want your cat and the outside cat to be aware of each other but not meet.  Once the cat has a clean bill of health you can swap scents using a clean cloth (wipe each cats checks w/ a different cloth and then let them smell the other cat), and site swapping.  If done right there is no reason the cats won't get along. 

 [article="32680"]How To Successfully Introduce Cats The Ultimate Guide​[/article]  

I would check the Petfbi for your city and also on craigslist to see if there is any adds. You can also place an add on both sites. 

As long as your cat is up to date with its shots, everything should be OK. Two males cats most likely won't get along. Any other combo should be fine. Best of luck and blessings!

Thanks to all for your advice!