How many vehicles does a typical american fire brigade employ on a regular fire suppression task, and how are they organised?
It entirely depends on the area, the event, and the available machines.
The standard American doctrine (I believe) is to over-respond and then redeploy if needed. So if there's a fire call, they send "everything" from the local station, and then if only a few trucks are needed the rest return (or divert to another call).
If it's medical, they usually send one. But again, this is entirely dependent on the local setup, how much they're covering, and what they can cover (and what further stations can cover them).
See
https://en.wikipedia.org/wiki/Multiple-alarm_fire (
a) for more details on some common practices. It's not uncommon at all for multiple stations to respond to an actual fire (which are pretty rare, all things considered, maybe 20% of all calls?) and then disperse once it's under control. Better to be there and not needed than needed and not there - it's not like the firemen have much to do besides polish the pole when they're not responding to a call.
The USA also sees wildfires that would blow European minds into the fucking stratosphere - the
August Complex Fire (
a) burned an area the size of South Ossetia - to which 2,900 firefighters responded. And that was a remote location! The 2007 fires displaced a million people and had 6,000 firefighters, not including 3,000 prisoners and military response. The
Thomas Fire (
a) had 8,500 firefighters deployed (largest in history for CA).
Europeans and New Yorkers can't even comprehend. When that many firefighters are deployed, they're pulling in equipment from other states and even other countries.
(That's an area the size of fucking Kazakstan or 1/10 the entire size of Europe)