It is a spell that is cast
The 3rd-level racial trait Duergar Magic states (SCAG, p. 104; emphasis mine):
you can cast the enlarge/reduce spell on yourself once with this trait, using only the spell’s enlarge option...You don’t need material components for either spell, and you can’t cast them while you’re in direct sunlight, although sunlight has no effect on them once cast.
Unfortunately, being in a Rage limits your spellcasting:
If you are able to cast spells, you can’t cast them or concentrate on them while raging.
To answer your specific questions:
- No, you can neither cast nor maintain concentration on a spell while raging.
- Therefore, once you started raging, it would end.
- It can be counterspelled because it's simply casting a spell. While it doesn't require material components, there are still verbal and somatic so it can be observed to be counterspelled.
- It does require concentration because Enlarge/Reduce requires it and nothing in the racial feature counteracts or supersedes that.