Would US police forces normally treat the use of a small amount of counterfeit money as an emergency?
This is much too broad to answer. However, if the police have no, or few, other incidents to deal with - why not? They’re there to prevent crime and a crime has been reported.
If someone called the emergency number (999) in Britain with such a report, I feel sure they would simply be told to try and get the vehicle registration number of the customer
In England (which is different to Britain), there have been significant cuts to police forces. I can say that not only do they not attend for the use of counterfeit money, they don’t attend many other things which you would want them to. I don’t think following the English model is the way to go here.
The other point to make is that the 911 call included the statement that the suspect was very drunk, and in (?) their car.
he was also drunk and everything and return to give us our cigarettes back and so he can, so he can go home but he doesn't want to do that, and he's sitting on his car cause he is awfully drunk
That is a good reason for police to be deployed, and probably to be arrested (I’m not sure about the rules in the US, but in England if you intend to drink drive you can be arrested to prevent it). I would expect a fairly quick response from police informed that a suspect intended to drive under the influence.
And, finally, the suspect was refusing to give back the cigarettes (possibly due to inebriation). That’s arguably theft, as the suspect paid with a counterfeit bill. To bring it back to the England comparisons, if you were to attempt pay for your groceries with counterfeit money, which the shop refused, yet you kept the groceries (possibly due to being inebriated), I think I would want police involved. Whether they would turn up is unfortunately down to if they can afford to.