Are chains locked to certain sectors? If so, is the a way of finding what sector you started a chain?

INFODUMP APPROACHING!

The Colonization chains only appear in each exploration cluster.

Ghosts of the Jem'Hadar, Project Chrysalis, the Consular Authority chains, the Colonial Team chains, and the first assignment of Children's Toys only appear in Cardassian space (Beta Ursae, Alpha Trianguli, Zeta Andromedae, and the attached exploration clusters). (I know Ghosts is restricted that way, not 100% sure of the others.)

For Culinary Credentials and Extreme Bartending, you have to visit your ship's interior and talk to the Chef and Bartender on the crew deck.

The Biochemical Investigations chains are all "Personal" assignments (they might show up at Department Head/ship interior contacts, not sure), so your current location shouldn't matter.**

Gaming Proficiency, AFAIK, is only available from your Operations officer (Department Head or ship interior), but location shouldn't matter on that either.**

Facility 4028 Fugitives seems to only appear in Federation (Sirius, Regulus, Pi Canis) and Klingon (Eta Eridani and Omega Leonis) space + exploration clusters.

** - I say shouldn't on these because the way the system is currently bugged, the Department Head assignments don't match the ship interior assignments. There may be additional bugs affecting where these show up, too, but logically they should not be dependent on location.