Can you name this Component and its function?

[Gaz64]

I see a carter low pressure pump drawing from the main tank which then supplies the tall alloy surge tank.
Your high pressure pump (not pictured) draws from the bottom of the surge tank to deliver fuel (AND no air) to the engine.
The return from the fuel regulator at the engine goes to the surge tank, overflow from the tank goes back to the main tank.

How long does it take to drop to 15 psi?

If it is virtually instantaneous, you have a weak high pressure pump or blocked filter issue.

If it takes some time, you have a weak or non-operating low pressure pump or pre-filter issue.

And as others have said, look for another shop if they can't work that out.