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No — you cannot play Fog of War cards during your turn; you may select them for initiative and the only normal way to remove them from the game is by taking a Recon action (see p.11 and p.13). Scout/Surveil/Navigate actions place Fog of War cards from your supply into your discard when you place control markers (p.12–13).

Yes — a soldier in a vehicle can perform support and combat card actions as normal and counts as being on the vehicle's tile, and may also use vehicle actions listed for the seat when you choose a vehicle action (disabled vehicles cannot perform vehicle actions) (p.10–11).

Hunker down returns that card to your supply and it is removed from your deck until you take a Bolster action to add it back; you cannot hunker down Fog of War cards (p.11), and Bolster lets you take cards from your supply into your discard (p.12).

No — you cannot take control of a tile that your opponent controls if they have a soldier on that tile; you may take control of an empty tile or a tile with an empty vehicle (p.13).

Place damage markers on a vehicle for each successful attack; when markers are equal to or above the vehicle's critical threshold it becomes disabled, and when markers equal its integrity it is destroyed (p.15). Repair removes a damage marker from a vehicle on the same tile (or the vehicle the soldier is in), and if markers fall below the critical threshold flip it back to ready; destroyed vehicles cannot be repaired (p.13).

The defender must remove a matching card if possible from their hand, otherwise from their discard pile, otherwise from their deck (shuffling afterwards); if none exist remove the combat counter from the board and remove all matching cards from the supply (p.15).