Tanks under 5-10gallons cannot cycle and build up beneficial bacteria so don't worry about that. For long term health you'll need to do a 100% water change every other day minimum. Preferably, perform a 100% every other day and do a 50% change on the days in between. Rinse the gravel and plants well to remove poop & bad bacteria.
I wouldn't worry too much about the "warts" or color under his gills. IMO it is all normal coloration. The color patterns you are seeing are very common in red bettas. Here are some pics of red bettas:
You've only had him for a very short period of time. Give him time to settle in and he'll warm up to you :)
Good luck, I'm glad he's ok!