1. The game is beginning. In the first turn, should I roll the die to move the climate marker, or I can skip this phrase at the first time?
You play out the climate phase. That is, you roll the die.
2. Player A is attacker, and player B is defender.
If player A lose the combat, he only spend 1 point movement or his dino will die and remove from the gameboard?
He has spend his movement (might in rare cases be more than 1 point), and is dead, that is removed from the board.