[3onthetree]

You would tear off the roofing where the new shed roof attaches to the existing, so you can flash old to new properly. Usually it can sit on top of the existing sheathing. Of course, after the existing structure, snow loads, wind loads, seismic loads, etc are addressed.

If you are just covering your existing patio out about 8' or so, there is not a huge run of slope to worry about. But you do have an existing skylight which limits the height of where the new ridge can attach. If you want more headroom, don't finish it with a ceiling. FYI a starting ballpark for those examples you show might be $25K.

 

[3onthetree]

Or, as an alternative...
https://patioroofriser.com/#

Interesting! (2) 3/8"x4" lag bolts into the rafter. I've had the power line cable pull out the lag bolts on the roof when a small branch has fallen on it.