For point two..
The norm for buying stock is to just register online with a major broker: Fidelity, Schwab,TD Ameritrade...etc, send them money to fund your purchase, make the stock purchase in your account, and then have a little faith. You could probably get them to physically transfer the stock certificates from them to you, but it is not the norm at all. I would plan on a fee being involved also.
The 10$ is for one trade... regardless of if you buy one share or many. So you wouldn't buy 1 share of a five dollar stock as your cost would be absurd. You might buy a hundred shares.