A. Just a sign of vomiting in a puppy can be caused by several things including parasites (typically roundworms or giardia), food intolerance (ate something he shouldn't have), or if it just happens once could be caused by stress or hyperactivity.
The gasping is a sign that some of the vomit has entered the trachea. This could be an incidental thing, or if it happens more than once could indicate a potential dangerous disorder called tracheoesophageal fistula (a channel that connects the trachea and esophagus). This is a rare disease however.
I would recommend having your puppy evaluated by a veterinarian with a good thorough physical exam and potential diagnostics to come up with a solid diagnosis so that appropriate treatment options are implemented.
A. That could be due to a gastrointestinal infection, inflammation, but foreign body ingestion and potentially obstruction should be considered as well. If he struggling to breath I would definitely suggest to take him to your veterinarian for a consultation as may require immediate medical attention. Otherwise a bland diet based on boiled chicken and rice given in small portions every 6 hours could be beneficial. No significant improvement in the next 12-24hours should be followed by a consultaton anyway and more investigation to understand the underlying cause.
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