Flag Bridges let the ship host a Task Force headquarters. Task Force bonuses only apply in the same system as their headquarters, but owing to risk and practicality most players use a single Task Force headquartered in their home system.
You may change Task Force HQ locations in the Task Force Organization (ctrl-f4) screen.
Damage control... is only kind of effective. 'In battle' repairs are typically only meaningful during long duration missile duels, or if both sides exhaust their missiles.