Baby kitten is sick! please answer!& help me with thiss!!please!!!!!(:?

alright so... about a week ago me & my friend were parked at the store and I turned around to see a man carrying a kitten to the road & throwing him into it, the turning around and going inside. Ok sooo the cat walks toward the store & toward me cause I'm calling out to it & I pick him up and he is the sweetest most lovable thing ever!! I swear!!!! I love him to death!! But ok so he had dirty matted up fur & he's was just scruffy & his tail looks gimpy & he was (& has still been!)- coughing,sneezing,runny nose, snot (sorry its gross, but yeah, its clear kinda maybe? But sometimes it straight blood lookin!!!!) When he sleeps he starts having to breath thru his mouth but like I wipe his nose constanlyyy!!! & he's eating and drinking well. He's also using the bathroom normally. I want to take him to the vet, the thingg is though, that we already have 2 cats here & one dog. Mama said I had about 2 weeks to find a home for this baby or else she would have to take him to the pound :( well now she's saying I can either choose to keep him & take him to the vet and everything & get rid of the other two (pound) or I can choose to keep my other 2. Well I'm not sure if she's actually going to make me give him up, cause idk buit I have reasons to think hat she'll let us keep him. :/ but the thing she's saying issss... If I want to take any to the vet(shots is the only thing my other 2 need) but that the one I choose to take to the vet is the one I'm keepin. We can only afford to take one but like I love them all :( & idk what to do?????????? So like will he get better & what does he have?????