[TABOB]

So my question is- do Epson-based printers (Epson 3880 in my case) push ink to the printhead from the cartridges, or is it pulled by dumpers/printhead?

This printer has a pressurized ink system (pump in the photo), so the answer is both.

During printing the printhead is generating vacuum, causing the one-way valves inside the dumpers to open and ink flows through.
During cleaning operation however, the capping station pump is doing the work, sucking ink through the nozzle plate.

 

[TABOB]

After I changed the lines and dampers- the ink is not coming to the dumpers. Since the lines are clear- I can see how it slowly moves toward the dumpers each time that I turn the printer on and the air pump kicks in.

Everything is probably OK
Just pull the ink trough the dumpers with a syringe until the dumpers are full, then put everything back and start the printer.

 

[TABOB]

Where is the pressurized air introduced into the system?

The pump is pressurizing the carts, and there is a bag inside containing the ink.
A good system because the pressure is kept stable regardless of ink amount or altitude, BUT when the pump fails the vacuum prevents ink flow.