Can a disk that wasn't activated by the Command Card still use the Battle Ability of that Command Card?

Is an embattled disk considered to be in two separate engagements? I.E. Does it both counter attack the unit pinning it and soon after attack the unit it is pinning (assuming it survives the first attack)?

The answer to both questions is yes. The rules don't say that a disk needs to be activated by a given command card to use the battle ability. In fact, the disk may already be previously activated. It does not need to be unpinned either. And, an embattled disk will apply its attack value if it survives the first engagement.

Disks using battle abilities do not have to activate to perform the ability, and can even use it while pinned.

Resolve a scrum one engagement at a time, starting at the top of the pinned disks (the one where the embattled disk is the defender).

If there is more than one embattled disk in a single scrum, players must resolve all engagements where an embattled disk is defending before the engagements where an embattled disk is attacking. This results in a top-down resolution, where disks not being pinned in the scrum attack first and each(*) engagement is resolved individually until none remain.

* When an embattled disk is removed as a casualty, allowing the disk on top of it to pin a disk that was below it a new engagement occurs but players cannot resolve the new engagement during the same melee phase.