No, it's not normal.
The vet can easily determine if it's kennel cough, but that wouldn't be persisting for several weeks -- she'd likely already be recovered.
Is she on heartworm prevention? Some areas of California have pretty high infection rates. Heartworm infected dogs make that gagging sound (like they're trying to clear a hairball). She's too young to test, so the HW test would be a waste of money -- dogs her age always test negative, but it can sometimes be a false negative (they're not too young to be infected--the antigen test just isn't reliable til they are between 6 mo. and 1 year old, as it only detects adult female heartworms, and the worms have to be 6-7 mo. old to reach maturity and show up on the test...at an earlier stage of infection, the juvenile worms are in there, but undetectable). Ask your vet what the clinical rate of HW infection is that he or she is seeing -- if it's zero, I wouldn't be concerned, but if it's showing up in the clinic, I would.
If she's not on HW prevention yet, I'd ask the vet to consider putting her on monthly Advantage Multi, as it appears to probably have the longest reach-back of any HW prevention product on the market -- a 2014 study in Veterinary Parasitology
(vol. 206, issues 1-2, 15 Nov. 2014, pp. 93-98) found keeping dogs on monthly Advantage Multi (plus Doxy) could kill immature adult worms 5 months after infection, so your dog is in the zone that it could help, if she's not already on HW prevention. She needs to be on some kind of prevention anyway, and if you are late getting started, this at least will help ensure she tests negative at your 1-year annual check up.
You might even share this article with your vet, if he or she is unfamiliar with the new research:
Experimental Dirofilaria immitis infection in dogs: Effects of doxycycline and Advantage Multi® administration on immature adult parasites
Last thing to check, does your pup ever regurgitate undigested food?