Sorry for the delay - I have had some real life issues demanding my attention. I will try and get a map up shortly.
Monsters as characters - The rules don't usually provide for it but it could be done. And there are some settings that have have information on some of these races - I think the Reavnica book based around a Magic The Gathering world has a goblin race. But I probably wouldn't allow it here for a variety of reasons. I also tend to discourage evil characters as a personal preference. But you can do a lot of things if the DM and group want to make it work.
The ledge - the flat ledge above you can be reached through some moderate climbing. The "path" of sorts that you are on goes all the way up to it. You can move up this terrain at half normal speed (difficult terrain ruls) all the way until you are on the ledge. No spider climb needed. But I would allow you to move up at a normal speed with spider climb. The spell essentially allows you to walk around on surfaces like a spider. You could walk on the ceiling of a room and just dangle there from your feet - with your hands completely free to do whatever. It would be just like walking on the floow only, well, upside down. So in this case - there is a vertical wall behind the shelf they are on (Note the pitcher posted recently) and if you got ot it you could simply walk straight up with hands free. It is a pretty cool spell. And if that didn't make it obvious- yes, your hands would be free to use a quarterstaff.
One other detail to note about this situation. The bugbears are on a ledge above you - you can look up and see the edge of the ledge, but not them unless the approach the edge like the one did when it threw down the log. They have cover while up there and would have partial cover (disadvantage on attacks) even at the edge. But there are also lots of trees and boulders where you are climbing up potentially granting those approaching the ledge some cover as well. As I said I will try to get a map up.