Is it feasible to install Sierra atop a Mountain Lion that won’t boot? My friend has a 17“ Macbook Pro, I think a 2010 model. 

It has had massive problems for a while: trivial finder operations that typically take fractions of a second (like opening a new window or changing a directory) would take minutes to complete; apps would randomly hang for a half hour.

There was clearly something amiss and I was going to save her data, wipe the drive and install a new OS. However, in the meantime my friend installed an OS X update (not Sierra, but a regular package with minor updates). 

Installing the update froze the computer for three days. The progress bar would move a little bit during the first day, then nothing. Once it was clear nothing was going to happen, we turned it off. Now, unsurprisingly, it won’t boot: you can see the apple and the OS’s progress bar; once that is loaded, the machine will reset.

I’d like to download Sierra and install it right away, but there’s important data on the drive, including an iTunes library that hasn’t been backed up in a long while. So it’s pretty important to try and build on the system’s old state.

Is there a chance the Sierra installer can fix the old system and do the upgrade, or do I need to find/download a DVD with the original system, and do a repair install first?
While your friend's MacBook Pro is compatible with macOS Sierra (MBP Mid-2010 or newer) I suggest to fix the issue first.

It sounds as if the HDD is corrupted. Try replacing the HDD or to boot from an external HDD (USB or FireWire).

Make sure everything runs smoothly before upgrading to a major version of OS X / macOS.