Buying means you buy already-minted bitcoins. You can buy them privately or on exchanges. Then they're yours. The exchanges' role is to facilitate trading between users, and they operate at a profit (they charge you a % to buy). Basically, it'd be hard to sell your bitcoin if you didn't have exchanges. You'd be posting on craigslist or something. And people would be scamming you for the money. Stuff like that.
Anyone can mine on virtually any hardware, however due to competition, it's hard to mine profitably. You have to have specialty equipment that is both very good and very efficient (power-wise). You can mine on your own or as part of a pool, but you essentially have to 'get lucky' on your own, and it's very unlikely that'll happen. My advice is to not think about mining until you've done a lot of research and for some reason decide you want to do it. It's a bad idea for almost every non-expert at this point.
Mining is based on a network agreed difficulty level. So as more miners participate and get better hardware the difficulty of the puzzle increases (by lowering the bar).
Currently it requires dedicated custom-built hardware (built solely for mining bitcoins) to make any kind of money mining Bitcoin. Even then you have little chance of winning the reward - so you normally would join a mining pool to pool computing power and share rewards with other miners.
The mining acts as a proof of work. It helps solves the problem that in a peer-peer system you can't trust anyone, because in a democratic system an attacker can run up any number of nodes they like at little cost. However with mining this is not possible, such nodes will not be trusted.
This is a very simplified view of things and there are more complexities but it gives you an idea.
But there's nothing within BTC itself that's volatile. The currency is slowly being inflated to the target 21 MM coins.
From page 3:
Nodes always consider the longest chain to be the correct one and will keep working on extending it. If two nodes broadcast different versions of the next block simultaneously, some nodes may receive one or the other first. In that case, they work on the first one they received, but save the other branch in case it becomes longer. The tie will be broken when the next proof-of-work is found and one branch becomes longer; the nodes that were working on the other branch will then switch to the longer one.