Wonderful to know that your upgrading! You're boys will be very happy
Here are a couple links that explain the cycling process, I had others but I don't know what happened to them....you can also google 'Aquarium cycle' or something of the sort
Personally, I think if you can, go with a fishless cycle. Heres a wonderful link that explains how to go about it
If your super edger to get your fish in there though, you could do a cycle with fish...but that can be a hassle for you could be doing partial water changes daily at times in order to keep your levels of ammonia and nitrite down.
Also, no matter what, you'll need a test kit..the liquid dropper kind. Those strips are inaccurate.
I also hear of some people using Seachem 'Stability' and a couple other products to assist in their cycling...I've never been able to find them around here so I've never tried it, hopefully someone who has will chime in
Probably silk plants would be more suited for you as opposed to live, all the beauty without the hassle of fertilizers and whatnot....and much easier to move, thats for sure.
Yes, they make small gravel vacs for smaller tanks and you should always have one...you'll be amazed how much waste you remove with them, even in a betta tank. Not to mention I find its far less stressful on the fish to syphon out water.
With a tank with a filter, I wouldn't suggest doing a full water change at all....you'll likely kill any good bacteria that forms. PWCs of 20%-50%(depending on take size)should do. I do weekly 20%-30% in all my tanks 10 gallons+ and then something around 30% twice a week for the 2 gallon. Smaller the tank, the more frequent the PWC.
Unfiltered tanks, IMO, should get PWCs every other day, removing any waste, and then a full clean once a week.
I prefer gravel, though some prefer sand. Its a personal preference really. I actually find gravel easier to deal with, but others will argue that sand is better.
Yes, I suggest you get a filter. There are many easy ways to modify the filter in order to make it safe for bettas and a filter means less water changes.
I'd say three is the max amount you could divide a 10....maybe four, but I think that would be seriously pushing it.
I hope I covered everything and I hope this helps!