Concerning staining of the woodwork, I suggest:
a) ask Laura to specify the stain she wants
b) purchase a quart (or a pint) of this stain
c) purchase a sample of trim (there may be some pieces in the garage)
d) experiment (in Belmont) to ascertain the result of staining wood that had been pre-treated with polyurethane, as compared with treating with polyurethane, wood that had been previously stained.  Then proceed accordingly on the Island.

Concerning the timing of the painting: The time of delivery (and painting of metal and wood) of the spiral stairs is an unknown variable, which might occasion a second trip by Tim, if he were willing. That second trip which might include the finish carpentry, might reasonably be delayed until the painting of walls and ceilings and the staining of wood had been completed.