Hi. I just started with armada and have a couple of questions about rules

On the second round, if all squadrons don't match the round initiative token and haven't been played in the ship phase, do you continue with moving or attacking with all of them?

yes.

Each turn has a particular color (you start with turn 1 being blue) and when you activate a squadron you toggle its slider to show the other color (so for turn 1, you set it to RED ).

You cannot proceed to round 2 until all squadrons have been activated and are showing RED.

On the second round, you continue moving and attacking with all of them, until they are showing BLUE.

Note that there are ways a squadron can get "toggled" earlier than the squadron phase: if this happens, it should not be showing the same color as the turn (so a squadron that got shot by dutch will be RED in turn 1) so you cannot activate it during the squadron phase.