If I took some gravel out of my cycled 10 gallon tank, as well as the water, and put it in a betta tank ......... would that tank then be considered 'cycled'?
Yes it will help with the gravel, you don't have any desesae in that tank do you, as i wouldn't bother if you have, as you will pass it onto the betta.
How big is the betta tank? If your not going to put a filter in it, then theres no point in putting the cycled gravel in there, because when you do a 100% water change you'll lose all the good bacteria.
You don't even need to take more than enough water to keep the gravel wet. Depending on the size of your tank and how long the tank's been running, it should work just fine. And, even if you don't have a filter in there - if you have an airstone running it will keep the good bacteria alive so you'll keep the cycle going.
The gravel will help cycle it but the water won't do any good. There is only a very miniscule amount of bacteria in the water column so you would be better off moving some gravel and using clean, dechlorinated water.