Let me just start with the basics--here is what you can do to use what you have to keep your fish healthy:
A 2 gallon container needs to have a 100% water change every 3-4 days. Don't worry about cycling until you have a Master Test Kit and know exactly what you are doing. This means you need to put him in his cup, dump all the water out of the tank, and clean the tank and everything in it with hot water from the tap. Once you've done that, you need to draw tap water that matches the temperature of the water in the betta's cup--preferably with a thermometer, but if you don't have one yet, you can dip your hand in the cup and then under the faucet until the temperatures feel the same. Fill up the tank, and add dechlorinator. Good dechlorinating products include Seachem Prime (my favorite) or if you cannot find that, Amquel+ is another good brand. Once you've added the dechlorinator let it sit for a few minutes and give it a good stir--then you can start acclimating your fish.
As far as acclimation goes, float his cup in the tank and every few minutes pour some of the old water out of his cup, and let some new in. If the new water is similar to the old (same source, same temp, same additives) do this over the course of about 15 minutes. If any of those components are different, do this over the course of 30-45 minutes. When the cup is full of the new water (you have to estimate) he is ready to be released into the tank.
You should not worry about the pH in your tank. Having the "perfect" pH is nowhere near as important as having a stable pH. When you add chemicals, the pH in the tank swings wildly, and this is where you run into issues with shocking the fish. It's better to simply allow the betta to adapt to whatever pH your water is normally than to mess with it. Bettas are fairly hardy fish and can adapt to harder water.
Your fish will need a heater. Bettas are tropical fish, and as cold blooded animals, they don't do well in temperatures outside of their comfort range (78-83 degrees F) lower temperatures can cause circulation problems, digestive problems, and a weakened immune system. Try to get your hands on a 25 watt heater with an adjustable temperature dial, such as this one: http://www.fosterandsmithaquatics.com/product /prod_display.cfm?c=3578+3743+11368&pcatid=11368
avoid pre-set heaters and heater pads. They are cheaper, but they are a low quality, useless waste of money.
If you think that the filter is bothering him, you can lessen the intensity of the bubbles by tying knots in the airline tubing so fewer bubbles are released at once. You may want to remove the plastic plant and replace it with a silk one. It's a good idea to supply your fish with ample silk plants for him to hide in and rest on, as well as at least one cave where he can hide and be in the dark when he feels stressed. A ceramic coffee mug or a piece of PVC pipe will make an excellent and cheap betta cave--just rinse thoroughly before use and make sure the items have not been washed with soap recently.
As far as the fin issues go--are you sure he has fin rot? Fin rot occurs when the fish's fins sustains damage either from being torn, or from the fish biting himself, or from ammonia being so high that it literally burns the delicate fin tissue. This injury then becomes infected with "fin rot." Fin rot has a very distinct appearance--the edge of the fin will be gray or black and usually has a crusty texture. If the injured edge of the fin is a normal color and texture or shows a clear or white outline, the fish is not infected. New finnage usually comes in white or clear and colors up later. This is often confused for fin rot.
Methylene blue is helpful for ammonia poisoning, but it can be hard to find in the store. This is part of why I order all my supples from Foster and Smith Aquatics online, lol. That, and the fact that the prices online typically run over $10 less than what you see in pet stores, which more than makes up for the cost of shipping if you order multiple items. I highly recommend doing your shopping there as opposed to a pet store--you'll save a ton of money and have access to a better variety of quality products.