US, yes. Yes, they do give chase or open fire with surprising frequency, often creating far more dangerous situations. 5 months average training time, additional training given through unions with a focus on "officer safety" (see, how to pull your gun before the other guy does) creates an ecosystem that creates a bunch of fucking cowboys rather than actual peace officers. Yes, a non-zero amount of them think they're playing cops and robbers for real.

It depends on where you are, the cop, and a lot of other context. It's one of those cases where America is more like 50 different little countries than one big country.

My state police force has a policy to only chase if there's an active danger to public safety.
That doesn't apply to the sheriff's of the 83 counties in the state, or the approximately 500 other police agencies, although many counties mirror the policies of the state police.

Weirdly, I generally trust the state police more than any of the others. They tend to be significantly better trained and more focused on public safety than making money for the county.
I've only been pulled over by one once and he just wanted to make sure I was okay, which was fair considering my car was failing and it sounded like a shitty old lawnmower that was also broken.

In general our police are powerfully undertrained, underpaid, over funded, improperly screened and with a radically unhealthy attitude on their relationship with non-police. We also lack enough uniformity for that assessment to be universal.

Would the chase not be more dangerous than whatever you are being fined for ?

It's not just cars, just recently cops in New York City shot each other and several bystanders on the subway trying to hit a guy with a knife they had been following over a <$3 fare evasion. Land of the free, home of the brave.