I guess the tricky part with the OP's plan is that he (I think) wants to bridge between the house wall and a wall on the boundary, so water run-off isn't as straightforward as adding guttering to a facia. You'd have to have some sort of channel/gutter on your side of the wall.
Yes this exactly, the wall of the house is straight and the garden wall is flat topped brick so about 8" wide. I'm thinking that I am going to have to install a channel on top of the wall to divert water to a butt/drain. If I have to use supporting columns then the space will be restricted sufficiently to scrap the idea.
I think you could probably put shed/greenhouse guttering on top of the crossways roof beams up against the boundary wall, with lengthways ones to support the roofing sheets -- if the lengthways bits are tall enough then the roof will end up sufficiently above the gutter. Depends on the boundary wall being tall enough, and indeed it being your wall
I'm 99% it's our wall, joined to our house, made of the same colour brick (next door being different) and is separated from next door by vehicular access to their parking.
Bear in mind that anything fully enclosed will count as part of your permitted development rights, so if you ever want an extension you've used up some of the size you could have had. Unless they've changed all the rules again, I can't be bothered to keep up
You're probably correct there, I may have to take the door off if we want to extend but it's probably only going to be 3-4sqm.