When I try to take the rear tire off, it feels like it is stuck on. No matter how much I loosen the bolt, the axle won't...

That is a Presta valve. Bikes often come with two types of valves, Presta and Schrader. Schrader valves are the same type of valve you find on car and truck tires. Higher end road and mountain bikes normally have Presta valves, while most entry level bikes have Schrader valves because Schrader valves are practically indestructible, and very easy to use; however, Schrader valves require a large hole through the rim, and are much bulkier than Presta valves. The pump you are trying to use is probably meant for Schrader valves only. You can either buy a new pump meant for Presta valves, or you can buy a small attachment, which will allow you to use the pump you have now.

Your shifting troubles is likely caused by your derailleurs. Dmadro's explanation is correct, but if you are unsure how to make these adjustments, you can take a look at one of the guides explaining each step on adjusting the front derailleur and rear derailleur.