If one is standing on the road as a bus approaches, can you signal for a pickup? Similarly, can you ask the driver to "let me off right here"? I'm asking specifically about Bus 962 from Donegal to Galway, but a general answer would be fine.
I expect to be staying about half-way between two rural stops (a 50 minute walk to both).
No, buses generally won't pick people up or drop them between stops unless the driver is in an extremely generous mood. Sorry to disappoint you.
A lot depends. I've found that rural bus services are a lot happier to do this than urban ones - especially dropping off. - if you talk nicely to the driver. I wouldn't guarantee them stopping though...
I would contact the operator, Bus Feda, to establish the correct position with regard to their specific policies- https://busfeda.ie/contact-us/
In the UK such a rural service would stop anywhere on request, advertised stop or not- but I can't speak for Ireland.