As they say, antibiotics are not for viral based illnesses.
Now, personally speaking:
1) my son had the Flu a couple of months ago. No antibiotics.
2) My son a couple of weeks ago, got sick with a cold that is just as you described per your son. It is a bug that is going around in my city. Lots of kids and adults are getting it. Now, my son is STILL coughing/mucus etc. So I took him to the Doctor. Because it was lingering. But his mucus is not green/yellow... so no antibiotics.
3) BUT... I myself, have gotten the SAME cold, that my son had. Same symptoms as your son. BUT, versus my son, MY mucus is Green. My getting that same cold, has progressed more, than in my son. AND I went to the Doctor yesterday... and I now have BRONCHITIS. AND so, I now have Antibiotics. And a nasal spray and an inhaler. All prescription.
And last week I also had Laryngitis as a result of that cold.
4) So the point is: everyone is different and the progression or seriousness of their cold. It MAY indeed warrant Antibiotics. For me, it is helping. I already notice a difference. And Bronchitis is contagious. As well.
5) Coughing occurs because: it is either Post-Nasal dripping and down the throat. Or, it is from the chest in origin. Both, create mucus.
But, if there is an infection, say a Sinus infection, then it CAN... infect the lungs too. I know... because I have had that happen to me. I now have Bronchitis.
So the other point is: you need to make sure, your son is getting and taking, what needs to be done. Even if that is Antibiotics. Because also, he is young. And you do not want his lungs... to get weak. It will then take longer for him, to recover.
AND, since he has had a Fever... that means, that he has/or had an infection of some kind.
Or that it was the Flu.
Flu... versus a "cold", includes Fever.
Also keep in mind, that young children often cough... when sick and when they have mucus... because they are not old enough to know how to SPIT out the mucus. And the mucus then accumulates in their throat or chest... and they can't get it out. And coughing will typically occur more at night... because the person is laying down, and therefore the mucus pools in the throat. And causes the gag-reflex and the person then coughs.
And your son may be throwing-up... because of the accumulated mucus, in his throat or stomach. Kids, when they cannot spit out mucus, they just swallow it... and then it goes in their stomach, for example.