1. Hehe, I've had problems with this one too. "01" is a slice number. To get rid of it quickly you can press Ctrl+H. But that will just hide it temporarily, so a better way is to go to "View" > "Show" and then unselect "Slices" (I think).
2. I tend to use "layer masks" to get rid of image backgrounds. It's a bit complicated to understand at first, but once you do, it is the easiest way! It means you can "paint" out the bits of the image that you don't want using almost whatever tool you want (brush, lasso + brush, pen tool, etc), and can let you do all kinds of spiffy effects.
Um, after doing a quick google search for "photoshop layer mask tutorial", this one seems pretty good:
http://www.lunacore.com/photoshop/tutorials/tut003.htm
You can probably skip to the second half if you want, that really shows masks in action, which is the best way of explaining them