There has been an adult cat, black and white with grey hairs in the black, hanging around my back door for several months. I've been feeding it trying to make friends. But everytime I feed it by hand - it bites my hand. Sharp teeth and its drawn blood 3 times. No, it wasn't a friendly bite.  It was more of a "why is your hand touching my food", bite.  I've tried not feeding it for a week but it still stuck around howling at the back doors (I have two).

It ain't going away. And he's made it clear he's not going to be a house cat, but winter is coming up.
 
See my dilemma?
 

Oh the poor thing! and your poor hand.(http://www.aesopsretreat.com/forum/Smileys/sarcasmics/shocked.gif) I'd keep a bowl of food and a bowl of water out for mr. Tom cat. And a bed. A blanky in a box, better yet a covered kitty house with a cozy blanket he can snuggle up in. Once you are friends..if he will eventually be:x I'd get him fixed so he won't make more strays. He thinks you are his new Dad lol. I don't know how an animal would survive in freezing winter weather so hopefully there's an enclosed porch or garage he could stay with a kitty door or maybe he could come in the house...

Don't give up on it, but wear tough gloves when you put food out.  It takes a long while for an abused or starving animal to build trust and if you call the shelter the cat will most likely get euthanized.

 
Yeah, I don't know if I should continue and hope for the best or stop feeding it and hope it looks for another home before actual winter sets in.
 
My wife, who is hooked to a breathing tube that followers her around, and with legs that get infected easily definitely does not want the cat in the house.
 
I don't know if I could build it an adequate shelter to keep it warm during the harshest below freezing weather. And I don't know any other way to make it go away without actually going outside and marking my territory on the porch and under the trailer where its been hiding from the rain. :- /
 

With the health issues, yes, the porch and under the trailer would be two good spots for Tom cat's shelter(s) -- He likes/is familiar with those areas so if there was an insulated, water-proof covered box for him with a warm blanket and food and water inside he might just get the idea these are his places.

If he's been here for months it looks like he's already made himself at home. I can't imagine how animals survive the freezing winter months outdoors other than mother nature gives them an extra thick coat of fur. With some insulated blanketed shelter and food and water I hope this guy makes it. 

If you have an animal society like Friends of Felines, which we have here you could contact them, rather than have the cat freezing outside when the weather gets worse, because it will bother you and certainly you can't risk having it in and your wife getting scratched.  The alternative is to build an outside shelter for it, but you do have a lot on your hands Jim and it's going to be an added chore.

Yes, it is a dilemma...and you're doing what you can.  I think you have three choices.  Borrow or rent a trap box (sometimes the shelters will let you use them to catch feral cats), and when it's caught, take it to the shelter...or get a simple shelter, made from a sturdy rubbermaid container, or one of those small igloo types dog houses, put in some old blankets, and keep it in a sheltered area, like pacifica said, under the porch, or near the garage and leave food and water inside; or stop feeding it and when it shows up at the door, shake a can full of coins at it to scare it off your property.  Eventually it will get the message. 

Don't let the cat scratch you again...the incidences of animal borne infections are too great, especially with feral animals...and I agree that it should never be brought into the house so long as someone has any kind of compromised immune or breathing issues.