Please help me with a feral/ stray cat

[LinhT]

I'm hoping to hear from the cat people who have experience with stray or feral cats. There is a (I think, adult) cat wandering around at night digging in my garbage, tearing open bags, pulling out pieces of spoiled food and leaving a trail. I saw it digging in my neighbor's garbage a few weeks ago then this morning I found garbage remains near my front door and my garbage bin was a mess, with ripped bags. I've seen a black cat with white paws and a white muzzle wandering around in my backyard lately.

I've never owned a kitty and know nothing about them because I am a dog person. Since winter is probably arriving soon in MN, I'm worried about the poor kitty. MN winters are no joke. I can't get close enough to it because it...
1) turns around and wanders back into the forest (no hissing or aggression so far)
2) I'm allergic to cats (I cough and have trouble breathing)

So I bought a bag of Nutrisource cat kibble today and put some out in the backyard. I'm not sure what else to do. Humane society can't really do anything since I don't have the animal in my possession. I doubt animal control would come out and comb through the forest in my backyard to look for one cat.

Should I even be feeding this little guy or girl?? Or would that do more harm than good? Is it even possible to "keep" an outdoor cat in an area where the winters can get -40 degrees F?

Even though I'm not a cat person, I would still be really bummed if I bring it to the Humane Society, only to find out it had to be euthanized due to whatever reason. Any thoughts, suggestions or ideas from the cat folks?