Do you find that when you follow a fat loss cutting diet that your strength levels also drop?

It's quite common for guys to lose strength when they cut calories and lose weight. Especially with the big compound power moves like the squat, bench, and deadlift. These moves are closely related to your bodyweight. For the most part when you gain bodyweight these lifts will increase and when you lose bodyweight they will decrease.

This is common among powerlifters. When they move up to a heavier weight class they will typically be stronger. And when they cut down to a lower weight class they'll generally lose some strength.

However, when it comes to bodyweight exercises such as push ups, pull ups, dips, etc. you will actually gain strength in those moves when you lose bodyweight because there is less weight for you to lift. If you lose 20 pounds of bodyweight then that's 20 pounds less that you have to lift.

Exercises that are geared towards muscular endurance will generally improve as you lose weight. Very often smaller exercises such as arms, shoulders, abs, calves, etc. will improve as you trim down and get in better overall condition.