Can you upgrade resupply depots?

Yes, with the correct research, the depots are upgradable. They can only supply a specific amount of supply/ordnance per turn to ships wandering around their system, as opposed to a full resupply when the ship stops at it's planet, but that's one of the values that increases as you upgrade the other facility. The other value that increases is just how much supply/ordnance they generage per turn.

1) Research 'Resupply' and you will get more 'levels' of the Resupply depot that produces more supply (and I believe distributes) per turn

2) Yes, so far I have seen that it works just fine (ships with low supply sitting at a warp point in a resupply depot system gain supply over time). You may not notice much of a difference if the ship carries a large amount of supply though, it will take time.

-Currently subjecting multiple galaxies to the doom of conquer or blackholes

Balance mod places Resupply Depots under the Resupply Tree. A couple hints for managing supplies:

1. The size of the planet your resupply depot matters. You want a big planet so that you'll build up a larger reserve over time.

2. Be careful when you upgrade supply containers... especially engines! Otherwise you'll wind up losing lots of supplies in the process because of the way those components are handled. Transfer the supplies over to the planet before retrofitting if it has room.

--
Supreme Chancellor Zannis of the T'Nar Republic

I've not run into a situation where I can make a significant dent in the supplies/ordnance levels on a planet, even with a basic level 1 depot. Of course, I'm still using the demo for this info, so I'm probably not getting as many ships. I routinely get to Light Crusiers by turn 100.

Mendel
Logic, my dear, merely enables one to be wrong with great authority