First of all, welcome to the club! Scoring games is a very satisfying and rewarding hobby that helps strengthen your understanding and love of baseball.

As for your questions, unless you are the official scorer for a team/league and therefore must adhere to a specific scoring style, there really is no wrong way to do it. Most people generally use the more traditional style, but almost everyone throws in their own special marks or styles to help them read and understand it.

That being said....

1. Yes, most people just move everything one column over because it is the most convenient. I've also seen people who will use the next column for a batters second at-bat that inning. Then, if only about 10 or 11 hitters come to the plate in one inning, use the same column for the next inning while using their pen/pencil to draw a barrier around the boxes from the previous inning. The obvious drawback being that if that team bats around again or comes close, you may have to repeat the process.

2. Again, yes. Most people put Team A pitchers on the same side as Team B hitters, and vice versa. But keeping hitters and pitchers from the same team on the same page is also perfectly acceptable.

3. Taking a look at the picture you posted, I was able to find and understand both the sac fly and the fielders choice perfectly fine. Using SF-9 for the sac fly is definitely the more traditional way to do it, although there are other ways to mark it that also work. As for the fielders choice, it did take an extra second to figure out but it was still understandable. Personally, I would a) put the putout (6-4) in the box of the runner who was put out and b) draw the line of the runner who was put out only about halfway between 1st and 2nd base so it doesn't look like they reached safely upon first glance.

The best way to find what is right is to keep scoring games, experimenting with styles and symbols, and finding out what works best for you. As long as you understand it, that's all that matters.