I’m doing something similar at the moment, albeit without the garden wall – I’m putting steel posts in. Polycarbonate roof, 2×4 rafters.
Slope it away from the house – that way if the gutter blocks it doesn’t soak your wall.
Check your upstairs windows can be cleaned from the inside – or, make a removable panel in the roof so you can poke a ladder up there every now and then. Think about how you’ll clean the new roof as well – they go green and benefit from a scrub once a year or so. Again, a removable panel or two might let you clean it from a ladder.
Not sure about flashing I’m afraid. I going to use some sort of adhesive/sealant type thing, but it’s against brick so reasonably flat. Personally I’d try to avoid cutting it into the stone as it’s a relatively temporary structure and the wall would look a mess if/when it’s removed.