First, I am glad to hear that her panting has settled. In this case and with its intermittent nature, I'd be most suspicious that it is a sign that she may feel uncomfortable with her GI signs (as opposed to a breathing or heart issue). As well, it is positive to hear she has a comfortable belly. As well, I cannot see your results of checking her gums, but hopefully they are properly pink as they should be. If these are all the case, then in regards ***** ***** question, this wouldn't be considered an emergency situation.
Instead, I do suspect that while her treats could be irritating her throat to cause her cough/gag, we more likely have an GI upset involving the stomach.This can certainly cause her vomiting, would depress the appetite due to nausea, and could cause those gassy gut sounds you heard.
With all this in mind, we can try Tookie on some supportive care just now. If you do hear any more excessive gas (which can distend the gut and be painful), you can treat her with Simethicone (which is over the counter at most pharmacies and often sold with human baby medications). When using this, you can give a dose of 1mg per pound of her body weight every 6-8 hour.
Further to this, to tackle that suspect nausea, we can also treat with an antacid. There are a number of antacids that are available over the counter and pet friendly. I would advise only treating with one, but the two I tend to use are:
*Pepcid (More Info/Dose @http://www.petplace.com/article/drug-library/library/over-the-counter/famotidine-pepcid)
*Zantac (More Info/Dose @ http://www.petplace.com/article/drug-library/library/over-the-counter/ranitidine-hcl-zantac)
These are usually given 20 minutes before offering food (to allow absorption) and of course you want to double check with your vet before use if Tookie has any pre-existing health issues or is on any medications you haven't mentioned.
Once that is on board, we can try her with a nice light diet. Examples of an easily digestible diet include cooked white rice with boiled chicken, boiled white fish, scrambled egg, or meat baby food (as long as its free from garlic or onion powder). Ideally, we want to offer this as small frequent meals to keep the stomach settled. As well, we'd want to give her a break from the chews for now, just in case they are irritating her throat or even triggering her nausea. And just to note, this will also help if her stools are a bit loose.
Overall, none of her signs sound suspicious of an emergency issue. Therefore, since she is resting comfortably, we can continue to let her do so and then start the above once she is awake. As long as she settles, we're happy. But if she is not 100% by Monday, then we'd want a wee check with her vet to make sure she hasn't picked up a GI bug or pancreatitis that may require further treatment.
All the best to you both,