Eurail Pass guidelines state that "reservations are required for all European night trains and most of the high-speed trains". Assuming availability, how far in advance must this be done? Can I just go to the ticket counter 10 minutes before the train departs?
How long is a piece of string? Popular trains at popular times fill up fast, less popular ones never do.
There's some variation across Europe, but reservations generally open 3 months before departure. And if there are still seats available, it's generally safe to assume you can purchase reserved seats up to 1 hour before departure at the ticket counter, although some companies/services have shorter cut-offs.
The longest such cut-off I know about is for TGV Ouigo (IIRC tickets must be purchased at least 4 hours before departure) but I am not sure that those trains are covered by your pass. Otherwise, buying a ticket or changing a seat reservation is typically still possible even a few minutes before departure.
But that's not what you should be concerned about, availability is. You did write "assuming availability" but limited availability is really what's special about those trains: a reservation is mandatory and there might even be a limited quota of seats for pass holders whereas you can board other trains without seat reservation.
Also, if you show up at the station 10 min before departure, you probably won't reach a ticket counter before the train has left.