Yes, if the darkness does not affect all the squares the creature occupies, the creature can still see from her non darkness squares.
Darkness is just a source of total concealment, which defines itself as the absence of line of sight.
Total Concealment: If you have line of effect to a target but not line of sight, he is considered to have total concealment ...
The word "you" would include equipment
The description might seem a bit vague, but 5E tends to treat "you" as being your character and whatever you are carrying.
As evidence, consider the time stop spell:
You briefly stop the flow of time for everyone but yourself. No time passes for other creatures, while you take 1d4 + 1 turns in a row,...
Light would continue to shine from the bullseye lantern like normal. The lantern would be rendered invisible as with anything else your carrying.
Being invisible doesn't make your gear transparent nor does it stop you from casting shadows.
Here's where its important to understand the spell description for invisibility:
A creature you touch becomes invisible ...
There are a number of ways to go about answering this sort of question, below are detailed the methods I usually use:
Specific Over General:
The two passages in question are (a):
"Anything the target is wearing or carrying is invisible as long as it is on the target’s person."
-- Invisibility Spell Description
"Until the spell ends,...
It’s up to the DM; however, it is likely that the Shadowfell is not pitch black all the time in 5e.
As you noted, the description of the Shadowfell includes the following:
It is a place of darkness that hates the light, where the sky is a
black vault with neither sun nor stars. (DMG pg 51)
For the purposes of this discussion, Domains of Dread and other ...