In many missions we need to scan items, vessels and more, but apparently not while moving. Let me give an example: In the Treasure Trading Station mission, we are asked to scan and disable 3 smugglers, but the scanning part can only be done while stationary. Never had the issue with the main character.

I also had this issue come up several times.

I had this issue also. Also in Treasure Trading the cargo vessels' statuses were switching between unregistered and ally and back again. I also couldn't see the hull damage part of the enemy icon at the top of my screen. It was blank, and would for very brief moments flash on, then off again.

Scanning allied freighters is impossible while in motion on both the Starbase 114 and Treasure Trading Station maps. I haven't had to do it on any others, so I don't know if it works elsewhere or not.

I've had the flickering target indicator problem on several maps, with Klingon, Gorn, Orion (probably), and freighter targets. It was especially bad on the Starbase 24 map when trying to find already-weakened targets to finish off.
To clarify, the information that comes up when you target an NPC entity will switch how it's displayed with no apparent cause. It goes from the <10km view (strength of shield facings, hull integrity, name of target, etc.) to the >10km view ("Enemy," no shield or hull information) and can rapidly switch back and forth in the middle of combat without the camera having been moved.

I just ran through the 'Treasure Trading Station' mission in a Tier 2 Escort, and I found I was unable the scan the freighters for contraband unless I dropped the ship to zero velocity...

I can confirm this. Also specify it to instance spaces but not Sector Space. I can scan Sector Space anomalies while moving (even though the rewards aren't hooked up). It's just in missions I need to stay still.