This sounds more like a gearbox that isn’t selecting fifth gear properly, rather than one where it’s leaping out of fifth. For it to pop out of gear when cruising along suggests it’s not getting into gear fully in the first place. You might be able to adjust the shifter’s position and alignment and therefore get fifth gear to engage more fully.

There’s also a chance that this refusal to select fifth gear is the result of bent or worn selector forks within the transmission itself, and that’s a gearbox-out job unfortunately. So the first step is to get underneath with a torch while somebody inside attempts to select fifth gear. If the selector linkage runs out of range of motion, you might just have an adjustment problem which can be pretty easily sorted.

Modern cars absolutely hate having low battery voltage. It throws everything into chaos and can easily produce the sort of problems you’re having as the major components of a car are, these days, all controlled by computers.

By turning the car off and on again, you’re effectively performing the old Alt-Ctrl-Del function to reboot a home computer or laptop. As such, and especially since the transmission works perfectly when you reset it, this is almost certainly an electrical problem. A more permanent reset might be needed, so try leaving the battery disconnected overnight and then reconnect it next morning. This will often return things to normal.

First things first: are you sure the leak is coming from the gearbox and not the engine? It can be very hard to discern the difference in a manual car, although it’s easier in an automatic as the transmission fluid is usually a red colour.

Beyond that, oil can escape from a gearbox through a variety of places. The breather could be allowing fluid out, the pan seal could be leaking, the rear gearbox seal could be faulty or you could even have a cracked transmission casing. Don’t rule out a leaking drain plug, either, as this is a very common source of a leak. You might even find the leak is coming from somewhere within the transmission cooler and its plumbing.