how do y'all give Baby medicine?

Right now my 14m son is dealing with a pretty nasty cold and cough. It's making him vomit pretty frequently and just feel miserable all day. He's exhausted from not getting sleep. I'm exhausted from washing mountains of vomit-covered vomit. We're trying to give a homeopathic cold and cough suppressant that comes sweetened. It's not a flavor (like grape or cherry) it's just sweet. We've tried droppers, the little cup with the nipple, and syringes. We've tried mixing it with water and milk. He doesn't drink juice (his choice) and isn't eating enough so I don't want to scare him off of what little bit he is eating by mixing meds & food. I'm lost! Could use some advice for this poor guy...

Edit** Just wanted to say thank you to everyone for all of your advice! We went to the doctor today and am starting antibiotics to combat an ear infection. This is the worst part of mommy hood! I'm just ready for my little monster to feel better.