Long story short - my players don```t have a void-master. So, how manoeuvre calculations will work? Just a crew modifier? How much stuff can crew do on the ship?

If you don't have any PC or NPC with Pilot (Space Craft) - then yes, you use crew rating. Since you have thousand of crew onboard, they can make any number of actions.

Also, anyone could pick up the relevant skill (via an elite advance if required) and stand in as the helmsman.

The standard rules are that the crew can attempt anything - I believe there's a recommendation that in combat they are allowed a number of extended actions equal to their crew rating bonus (so 3 for a crew rating of 30, for example) - someone will have the respective skill.

Why not make some NPC officers?

Where were the crew roles and their effects given - Into the Storm?