From: Boston, Mass
Almost everything here has been answered to some extent, but some are not clear, and for new readers, I'll put them all in one post.
1. Ships can be re-supplied at sea from ANY HQ with cargo capacity. Just like strategic HQ-HQ supply transfers on land. NOTE, the cargoships MUST be part of the HQ SF, not just present in the port.
2. HQ's being transported are inactive units, just like any other. No Supplies recieved, none delivered, no production received, no combat bonus. Sometimes this can be convenient, as you can transport a tiny HQ, and have it collect all the stored prodcution as soon as it disembarks (providing it survives that first turn).
3. SEA and AIR units do not get combat bonuses from HQ's
4. When SEA units dock at ANY port (HQ or not), they attempt to draw supply from their parent HQ, wherever it is. This can be a HUGE drawdown, so beware of the size of your sea formations if you end the turn in port. Change their HQ if necessary to get the best supply distribution if you can.
5. Carrier SF's are a single unit, unlike cargo ships with loaded units. Any air action includes ALL the aircraft. Note to #4, Carriers can suck up vast quantities of supply.
6. ON larger maps, carriers make great island airbases. Fly (transfer) your 5-n planes from carrier to destination land base, fly (transfer)new planes from source base to carrier. Repeat. (note to self, DON'T get it sunk). This may not be such a great idea with the new naval AI in v2.14
7. Production can go from a port (connected) city to an HQ connected via a port, subject to distance rules, similar to the land supply line rules. If I remember right, there do not have to be cargoships in the ports, but it helps if you need to move more stuff as well. BIG BUT though, any transfer of this sort is subject to interdiction by both air and naval units along the supply line.
8. Note to #7, this happens automatically, and at a huge disadvantage if you have no convenient ports, to any unit being supplied from an HQ on a different land mass.
9. Produced sea units arrive in the port creating them, but they belong to the HQ to which the production was directed. If that HQ has no supply (to the port), the ships will have NO AP when they arrive.
10. Off topic to #7. This is true of aircraft as well. They will be delivered anywhere they can reach and will NOT be interdicted by naval forces; I'm not so sure about air interception.
11. SeaCap works just like land Cap (rail/truck). Its simply a strategic transfer of a unit or SF from one SF to another. Like #1, cargoships must be part of the HQ SF.