Can you count the cards left in the deck?

Yes, on your turn.

It seems like you can cheat when Evading, Mobilizing, or responding to Attack/Sabotage/Probe.

We suggest that you don't cheat and don't play with cheaters.

On the Dynasty Launch Fleet mission does the phase "up to 3 cities away" apply to both Occupy and Attack?

For Launch Fleet, the "up to 3 cities away" does indeed apply to both Occupy and Attack.

On the Liberation mission Mobilize Base are the cities named, excluding your base, restricted to cities in the Liberation player's hand?

For Mobilize Base, the other 2 cities can be any cities you want. (Of course you should choose them strategically!)

What happens if you Recruit Spy with fewer than 3 cards in the deck?

You look at as many cards as there are, and do not reshuffle the deck