Been feeding a stray cat on my block... Not sure what I should do. Any suggestions?

We just moved to the area a month ago. The day we moved in a BEAUTIFUL & healthy male cat came running up to me. He was SUPER friendly. He jumped right in my arms and was rubbing, licking & purring all over me! He was adorable! I figured he must belong to someone in my neighborhood because he is very friendly. He even tries to come inside my house. He will stand and paw at my front door. So I started to put a bowl of food near my front door for him everyday. He seems to go from house to house it looks like. He looks pretty healthy for a stray cat but he has no collar. He is also intact & I would hate for him to make little babies & add to the stray animal population. He comes to my house everyday ritually. The woman who used to live here had a few dogs & cats so I wonder if maybe she used to let him inside. I would let him in but I already have 4 cats and a dog. My husband said we can't keep him cause we have enough pets PLUS we are renting this place & it is mostly carpet. I heard about programs that neuter stray cats for free & then return them to where they found them. Is that true? I am tempted to keep him but he seems so happy being outdoors & I wonder if it would be hard to train him to be an indoor cat. Plus, I am really not certain if he may belong to someone in my area. I just feel bad cause the other night he came into my garage meowing, all wet & cold cause it was raining hard all day & chilly outside. If he does belong to someone, I don't know how anyone could do that to their cat & be so irresponsible to let an intact male cat roam around the neighborhood! Sorry, for the long story but I needed to vent. Thanks for listening.