KITTY KITTY

So I need some advise on what to flippen do here. So last night we came home to a cat meowing at our gate so we all petted the cat and then went inside to get ready for bed. Then this morning we woke up to the cat meowing on top of the girls playhouse out on the side of our yard. So I called Greg and told him that the cat was still here and asked him what we should do because he acts like he is staving. So I feed the cat some ham that we had and some bologna and he gobbled it right up. So all day today he has been hanging out at our house and meowing. He is a really cute cat and he loves the kids especially Brooklyn. Brooklyn was outside for over an hour playing with him and holding him like a baby and the cat didn't even seem to mind in fact if Brooklyn would walk away the cat would run right after her for more attention. The kids have been wanting a cat for awhile now and Greg and I have been putting it off for a lot of reasons. One of the reasons why I didn't want a cat right now is because I don't want a cat living in my house it just grosses me out and this cat seems like he lived inside because if someone is not outside with him he meows. I really think that someone just dumped him off in our neighborhood because he doesn't have a collar and he is just a sweetheart. So my question to you guys is do we keep the poor little cat that doesn't have a home or do any of you want a great cat that is great with kids and would be able to be inside because if he lives at my house he will be an outdoor cat and it gets pretty hot here in Phoenix in the summer. So write me and tell me what I should do here.