I don't see regrowth.. I see massive fin rot that is verging on body rot.. I'd treat that with antibiotics but if you don't want to you can treat conservatively with aquarium salt (predissolved) see this sticky here: http://www.bettafish.com/showthread.php?t=73332
Nice upgrade :) 10 gallon tanks are much easier to maintain.
Please have a read through this thread of cycling http://www.bettafish.com/showthread.php?t=47838
and this thread of water maintenance http://www.bettafish.com/showthread.php?t=66595
(once the tank is cycled which could take up to 8 weeks).
What kind of water conditioner are you using?
You're probably going to want to turn that heater diagonal across the back so you can fully submerge it, you also don't want to rest it on anything like the gravel. Use the clips include it and stick it on the back.
You should turn the lights off at night so your betta can sleep. They need darkness to sleep.
I suggest putting an intake/prefilter on the aquarium because they can be sucked in. You may also need to baffle it.
Those stick on thermometers are really inaccurate. I would get a glass in tank thermometer with the suction cup that sticks it to the outside.