Does Teleport Strike Count as Moving?

Do terminators shoot their heavy weapons with a -1 penalty the turn in which they arrive on the board due to teleport strike (counting as having moved)? Thanks! Yes.

Does deep strike count as moving?

Units that arrive as reinforcements count as having moved in their Movement phase for all rules purposes, such as shooting Heavy weapons. Any unit that has not arrived on the battlefield by the end of the battle counts as having been destroyed.

Can Terminator Teleport strikes Turn 1?

Nothing in "Death From Above" or "Teleport Strike" rules deny them from coming down on turn 1. Except the matched play rule, that would also limit how Drop Pods can arrive.