1 Remotely query the deployed boxes and pull the results back to the homebase box for processing
A standard connection with SSL enabled and 'forced' would provide what you're looking for (see the answer for #4).
2 Nightly (remote) "sync" or dump the deployed boxes' databases to a master database on the homebase box. Although each remote has box has the same table layout, the actual data differs across the board. Being able to dump all of this data remotely into master DB would be ideal.
3 Remotely push a table entry to all of the deployed boxes from the homebase box.
Have you considered one of the many replication solutions available, including Slony?
4 Ensure security of data in transit, and remotely deployed boxes
PostgreSQL supports SSL out-of-box. You can also force each server instance to require SSL for connections. This will provide encryption at the transport level. I believe newer versions also support client certificates, if you want to have an additional layer of security.