My dog bites the hand that feeds him

I am in a bit of a pickle with this dog.  Although he is sweet and gets along with the kids, he has almost enough issues to make me rehome him.  We found him on the street and I understand he is going to be aggressive with his food because of it.  but it's like he is bipolar. one minute he is lovey dovey, the next thing you know he chases the cat til she nearly pee's herself for no reason.  I am sure she is thrilled at the fact that we now have a dog that chases her all the time. Poor thing.  So last night Hayleigh was trying to give the cat a treat, then like a bold of lightening, Fedor races over to where she is and starts biting and growling at her hands.  Apparently he thought those were his treats.  Either way, you bite my kids you are going down.  I just don't know if I am acting too soon and not giving him a chance.