Hi Aria! Welcome to bettas! I am so glad that you want to do what's best for your new little guy, and this site will be a great resource for you. It certainly has been for me!
To address a couple of your questions:
You have a 1 gallon tank, so you will not want to try to cycle that. You need to do a 50% and a 100% water change minimum
per week. I have a 1.5 that I opt to do 100% every 3-4 days. Otherwise the ammonia level in your bowl will get too high for your fish. I would also recommend rinsing your rocks with hot water and scrubbing them off with your fingers to remove that slime. I use the cup my fish came in to hold him while I clean his tank, and since he comes up to meet me I can usually scoop him without even using a net.
Be sure you put a top on any container you keep your fish in because bettas have been known to jump.
He would also probably enjoy some silk plants and some kind of cave to hide in/explore. It's great that you are using a dechlorinator in your water before you add your betta; that is very important as well. I would not bother with the bacteria additive, though because you are not going to establish a cycle in your bowl, so it is a waste.
I would not rush to treat him for fin rot if his fins look healthy. I would just clean his water and bowl and watch at this point. He could just be producing excess slime coat. If it gets worse, or he starts to look/act ill you can post in the fish illness section on the boards here, and some terrific experts will guide you in treatment options.
With a room temp of only 70-75, you will need to consider getting him a small heater. Bettas like their water to be near 78, and lower temps can compromise their immune systems.
If you get a larger tank for him, no you should not try to add a female. They will
fight, and it is likely that one or both would be seriously injured or killed.
Transporting your betta 40 min. will be pretty easy. If you brought him home in a cup, just put him in there for the ride. With warm summer weather, you should not have to worry too much about temp, but if the car will be cold, you can wrap the cup in a towel to retain heat. That is not a very long trip, so he should be perfectly fine.
Bettas like to have some down time, so I would use the lights during the day/evening and leave them off at night. I have LED lights on three of my tanks, and they seem to be fine. The fish do have places to hide and get out of the light if they want to, though - plants, etc. Personally, I prefer fluorescent lighting just because it tends to bring out the colors of the fish more. LED's tend to wash out some of the lighter colors in my opinion. They have an advantage, though, in that they do not mess with temperature and do not require bulb changes.
I'm not really sure about a good acrylic starter kit that would be that cheap. Acrylic tends to be a little higher priced than glass. I got my 3 gal. acrylic kit from Petsmart for about $40. It is a Marineland Crescent 3. I like it, but it does have some quirks. You have to remove the light to do anything because it sits close to the top of the tank, and the lid is a little finicky. However, it is a bowfront, so you have a nice view of your fish, and the filter is pretty easy to reduce the flow on.
Anyway, I hope I have been able to help answer some of your questions. Good luck with your new friend! Bettas are the best!