Planning my Bathroom Remodel
You can drill holes in the existing joists to accommodate the new plumbing. The holes cannot exceed 1/3 the depth of the joist, which is about 2-1/2" on an 8" joist. The holes can't be within 2" of the bottom or 2" of the top of the joist, but can be located anywhere along the length of the joist. The plumber will drill the holes and piece in the PVC drain in the floor. You won't have to do it for him.
Some plumbers are quick to notch wood joists. This can be done, but it substantially weakens the structure. Notching is allowed in the joist, but never in the middle third of the joist's span. Notches can't exceed 1/3 of the joist's depth.
Just go over his plan before he starts to make sure he doesn't do any wood butchering that will compromise the structure or cause you to fail your inspection.
As for costs, that's not what this site is for. Guessing what it would cost you would be a shot in the dark. We're better geared to helping you do these things yourself.