impressive maneuver -
that turn in the drain arm calls for a long radius 90, but should be ok, since it is a sink...
anything that isn't buried in the ground needs a shielded coupler.
they make them material specific - so 1.25cu to 1.25pvc (they make 22.5 and 30 deg pvc short bends too)
alternately, sweat a female cu threaded on, and use male pvc.
scavenge more pvc from somewhere, and rip out the copper for $$$. !!! more bike bits.
(ie, tie vents together in wall, sell copper, make less holes in roof)
for bending connections, use heavy duty tigerflex. can internet in short lengths. PVC shell welds to std fittings
see if a J hanger will attach the stub-out to the stud, otherwise a piece of wood coming up under it to maintain slope.
maybe some rigid insulation up in the corner of the tub, against the outside wall (above rafter?? is is outside?)
coming along nicely, niche envy.