I am more than happy to go into this issue in depth, but please appreciate that I have to always give the safest possible advice for your dog and in this case because of a range of symptoms which are giving me cause for concern such as the stumbling and the panting I would recommend that you have your dog checked over sooner rather than later. If I was to give people home remedies and it was a serious disease the consequences could be dire.
Having said that I can address some of the issues for you ...
1. This does not sound like a twisted stomach, the signs of this would be a very bloated abdomen which was getting worse by the minute. But as this is a rapidly fatal disease you would have to seek immediate help.
2. It could be something stuck in the throat, this would be rare but again requires immediate attention.
3. Some respiratory diseases like kennel cough
can make the owners think there is something stuck in the throat and they can throw up sputum which can be foamy as well. While this is at trivial disease it does respond well to antibiotics.
4. The licking of the paws does not fit in with any of the above but could be stress or allergic skin disease.
I am happy to try and help you so if you have any more specific questions or want further explanation please come straight back to me.