Keep in mind:
Max Health = Health + Shield
Health can be from 1 - 100
Shield can be from 0 - 100
Mini Shield gives you +25 Shield until you are at 50 Shield.
Big Shield gives you +50 up to 100 Shield
AR (assault rifle) deals 33 damage
A Purple Shotgun deals 30 - 200 damage (in this scenario 110 damage)
A Gold Rocket deals 130 damage
A Cracked shield means you have 0 Shield remaining

USE SLOW MOTION

Pause at 4:04. What is the max possible health of the other player?

70 HP
Max Health is 200. The rocket does 130 damage, so the most Max Health they
could have now, after being hit by the rocket is 70, because 200 - 130 is 70.

How many times would Jesse have to hit them with a shotgun that does 90 damage?

Once
If the player has less than 90 health, then he can “one shot” them,
because 70 - 90 < 0.