Need timber advice for building an outdoor frame to support a roof
Pine is fine.  I'd treat it with a preservative and I do prefer to do this on the ground on sawhorses.  However depending on the preservative you use, you may have to handle the wood with rubber gloves.  Be sure to read the instructions carefully.  

I'd cut the wood myself for sure.  Many times its better to cut by putting the material in place and marking rather than measuring.  Also, it's a rare lumber yard that I'd trust to cut everything accurately.   

If you have a good impact driver or drill, I'd recommend using the special screws they make for with the sheet metal hanger brackets, just easier.  

Make sure those posts are sturdy enough for the job.  

Your drawing does not show a ledger - a board fastened to the wall to which you'll attach the rafters using metal hangers.  The secure attachment of the ledger to the building is very important for safety.  Usually you can't count on anchors into brick because you can't be sure of the strength of the brick.  

Your building codes may require that the ledger be attached by bolts or large screws through the brick into the framing on the interior.  If this is too difficult or impractical, you may be able to add posts under the ledger at the wall, so the posts carry the weight of the roof.