Correction

A question

Yeaterday Yesterday, I hang out (hung out [past tense]) with my brother, still he got a little cold and didn't feel very well. He just don't (didn't [past tense]) want to stay at home.The point is, I knew he is (was [past tense]) ill and I (still) share(d) a bottle of water with he him. And (today /or/ now), I have a runny nose today...
If someone got (gets [past tense]) influenza in summer, can we call it "have a cold?"

A question
Yesterday, I hung out with my brother, he got a little cold and didn't feel very well. He just didn't want to stay at home.The point is, I knew he was ill and I still shared a bottle of water with him. Now, I have a runny nose today...
If someone gets influenza in summer, can we call it "a cold?"

**The only mistakes were just with having the wrong tense's. present >>>past. "have a cold?" >>>>It doesn't sound natural to say it this way. In your context, you don't need to say have a cold, since you have already stated before that someone has something. Good work