My friend gave me a beta in a small bowl (with fake plant and gravel). I read into beta care and got a used 10 Gal tank that came with thermometer, used Elite Radiant Heater & used Tetra Filter. Cleaned them all with water only (thoroughly) and bought a new filter cartridge.

DAY 1
1. The 10 gal tank is filled 3/4 as I was concern with the weight of the tank on our office table. Used tap water with Beta Water Conditioner (Top Fin Brand).
2. Kept the temperature is at 75-80F
3. Ran for 24 hrs without the fish

DAY 2
4. Transferred the gravel from the original bowl (after washing them with water) into the new tank.
5. Transferred the Beta into the new tank in a ziplock bag with its old water. Let the bag float in the new tank for 2 hrs before releasing the Beta
6. Water looks fine, Beta is happy, swimming around.

DAY 3
7. The filter was turned off overnight! Came to work with cloudy water (white).
8. Temp is still between 75-80F: Heater is working well.
9. Betta still looks fine.

CLOUDY WATER: I'm wondering if it was transferring the old gravel & water from the Beta's bowl into the new tank and not running for 24hrs caused it, OR the filter was turned off last night?

QUESTION: Should I do 10% water change right now but keep it at the current 7.5gallon? OR Just add more water to fill the 10gallon? OR Buy brand new filter? OR just wait for another day? What are my options since I cannot do a water test right now.

Thanks!

Sounds like you've done everything right.
Anytime you have a new (or used) aquarium and filter it has to go through a process called the nitrogen cycle or the aquarium cycle and this causes cloudy water. The filter should remain running at all times so that this process finishes, never over clean the tank, change filter cartridges and make water changes on different days.

For now, you can just leave things the way they are and keep food to a minimum. The water will clear on it's own, may take a few weeks, but it will clear.