An enemy fleet is en route from a warp gate that he owns to a warp gate that the defender owns. The defender destroys his warp gate and the enemy fleet slows down to normal speed. If the defender then builds a new warp gate on his star, will the attacking fleet regain its original warp speed. I would assume not but would like some clarification on this.
That is correct, the fleet will regain its warp speed once the gate is rebuilt. Sometimes this doesn’t show correctly when a gate is built, but a refresh of the browser usually works.
Yes, if you really want to mess the enemy over, build a gate on a nearby star, use it to jump to the enemy’s star and destroy his gate as well. Then if you need to rebuild your own gate on the original star, you can do so without speeding up the enemy fleet.
Another tactic you might consider using, though it counts on you having disposable income, is to build a gate on your own star that is under attack, allowing you to jump your ships closer than incoming enemy ships. Then destroy the receiving gate before your enemy can utilize the bonus. Make sure you can see the star that the enemy launched from, just to make sure they don’t already have a gate.
I had a game where my enemy would launch to my gated star, I would shut down the gate at that star, forcing a 12h trip with gates to turn into a 36h trip. Then find the next closest star, gate that, and fling my fleets forward to that star, getting to the original star in more than enough time for the enemy to hopelessly suicide into my fleets 
It was FUN.