Yea, there are some stocks that I wish I had bought 1 share back in the day. BRK.A comes to mind. Hmmm, 10K account. It's pretty easy to blow through that quickly. I was enrolled in a day trading program, pretty good one, I think. I had a 6K account, and a trader to watch my trades and help keep me from making stupid mistakes. I still managed to blow through that 6K in about six weeks. Be really careful with that. You don't have much room for error. It's good that you're not on margin. Most don't consider the risks.
Yes. Although they sell your orderflow to the highest bidder. If you only want to buy 1 share at a time, it seems like an obvious choice. Or other places that give some free commissions based on account size (Merrill Edge comes to mind).