How worried should I be about this cough?

My dog began showing symptoms by throwing up; after that, she’s made horrible hacking noises every so often like she wants to throw up but can’t. I think this is a "cough"? It doesn’t sound like my friend’s dog who used to cough a lot, which was more of a dry wheeze. When my dog coughs, sometimes white slime comes up, but not much (maybe a teaspoon). The rest of the time nothing comes up; she’s only thrown up one other time, and it was after I gave her a bunch of dairy and then played roughly with her, so I am pretty sure that one was my fault. Is this a "wet cough" which I am told is very dangerous? Or is this "retching" or something else instead?

Also, in this situation, how soon is okay to go to the vet? She has her first wellness visit in 2 days, is this an emergency room situation or a "see your vet this week" situation? We assume this is a "don’t over-exercise and keep warm" situation in case it’s a virus of some kind?

Added as per comments: so over the last few days she’s had several different noises. The one that initially worried me is more of a wet hacking, occasionally with some phlegm or something spit up, and I’ve also seen her hack become full-on vomit. But this video captures the most common noise, a dry cough that she gets reliably after exerting herself (here by playing her favorite game, tug): https://www.youtube.com/watch?v=zCXJcmoTJNg (ignore the growling, she always growls during tug, it’s the coughing we’re looking at).