A warrior is in melee combat with an enemy and dying, and still has his ability "Action Surge" Warrior.
1) He can use the "Disengage" action to stray from 6 squares (your speed) and the Action appears to simply walk over 6 square?
2) He can use the "Disengage" action to stray from 6 squares (your speed) and the Action appears to use a Dash Action, and walk with this action 6 more squares?
Using Action Surge, he can Disengage and move 60 feet (12 squares) total:
1. Regular action to Disengage. For the rest of the turn, the fighter does not provoke opportunity attacks by moving.
2. Regular movement for 30 feet (6 squares).
3. Action Surge to Dash, and move another 30 feet (6 squares).
If the fighter is willing to risk opportunity attacks, he can move 90 feet (18 squares):
1. Regular action to Dash, and move 30 feet (6 squares).
2. Regular movement for 30 feet (6 squares).
3. Action Surge to Dash, and move another 30 feet (6 squares).
Note that if the enemy is melee-only and has a 30-foot movement rate, the second approach is often better. If you Disengage-and-Dash, the enemy can Dash and catch up to you. Now you're in the same situation as before, but you've used up your Action Surge.
If you Dash-and-Dash, you'll provoke one opportunity attack; but if you survive, you'll be 90 feet away. On its turn, the enemy can Dash and reduce that gap to 30 feet, but then you Dash and open it up to 90 again. The enemy can't ever catch up to you as long as you keep Dashing.