Will drives usually stop spinning when not used,
Drives can stop spinning when they are not used.
How soon that happens depends on the configuration of both the drive and the OS. The OS because it can send the drive a command to spin down. Or on the drive since modern SATA drives have the capability to do this on their own.
You did mention an OS, but if you run a unix (e.g. a BSD or a Linux distribution) then look at
hdparm. This is a persistent setting on the drive. (In other words: you only have to set it once).
Note that regular drive access (such as flushing log files or writing to the event log) will reset this timer. So it works best if you do not use the drive to host the OS.
This can best be combined with a small SSD (say 40GB-ish) which will not spin nor make noise. The SSD then holds the OS. The large spinning disk the data.
and the heat then so low that the fan will not be working?
The heat from drives usually is not a problem unless you run some old 15K RPM high performance drives. For a HTPC you would use a low RPM green drive. These are notably slower when doing random access (e.g. when using an OS or when running programs), but their sequential read/write speed is more than enough for movies, music etc.
As for the silent part of a drive: Get a low RPM drive. Usually those are branded as 'green'.
Not asked but: Why an external drive?
Putting the storage drive inside the HTPC would have less clutter and would be faster. (Navtive SATA access rather than using an extra SATA to USB part, an extra case and an extra power supply).