As has been said - a lot depends on the airline/FMS/SOPs...
For me:
1. Who initializes and sets up a route (FO, PF, ...)?
FO - while the Captain does the outside check (and yes - only the captain does it - even when it rains, snows, the wind is howling, ...
)
2. How do you check SID and the route?
I either enter the SID along with the route or wait for the clearance. At our FMS to enter the SID takes about 10-30 seconds, depending on your speed.
Once I'm finished I usually copy the Flightplan to the captains FMS. He will check it against the OFP before we do the "Crew at Station" checklist.
3. Who enters and executes the changes during the flight?
Autopilot still off or u/s: PNF on order from PF. Autopilot engaged: PF
Take off data is done by the PF, more often though both look at the charts together.