Hi Pan Marek,
Yes to answer your question, officers/commanders get their own unit card but are treated slightly differently from 'normal' units. They count as a 'heroic' type figure who doesn't get a Class or Morale factor but get a saving throw if they're hit by fire, either individually or as part of a unit that has suffered casualties. This makes them a bit tougher than they obviously would be in real life but is part of the 'heroic' focus of StEA. They can be equipped as you wish, but normally I would treat them as armed with a pistol, unless you were running a particularly 'pulpy' game and wanted them to play an outsized role.
HMGs and guns get their own cards with normal attributes of Class and Morale factor (C3 M3, regular etc).
Officers/commanders can move independently on their own action card, so they can shoot and engage in melee and, without a copy of the rules to hand, I believe they can also try and spot hidden enemy units.
When they are attached to a unit, as you say the unit moves on its own action card, and the officer's movement bonus is added to their move, +3" for infantry and foot-borne heavy weapons, +5" for cavalry and vehicles. So a unit of infantry that rolls 2D6 and gets a 7" move would then add 3" for the officer, giving them a total move that turn of 10".
Does that answer your questions?