You need to start the GTS server, not the GDS server to distribute Pokemon. The GDS is an additional program to allow reservations to be distributed out via the GTS.
EDIT:
1) You do not need to pay for this, although I wouldn't mind a donation
2) It can stay on as long as you want it to. I ran the program for nearly three days straight without issues.
To successfully distribute a Pokemon, you need to have the DNS and GTS servers running. The GDS is optional.