Well I'm not going to act like I'm an expert on Brinstar, because I'm not. I know Kage likes that stage so hopefully he'll give you some advice. I'd imagine that you'd usually want to stay near the center of the stage (as long as it's safe) and space attacks to keep your opponents offstage/at bay.
As for the ledge dash, the key is being able to jump IMMEDIATELY after you drop from the ledge. This will give you the height you need to be on top of the stage by the time you hit R (or L). There are basically 2 main methods used for the ledge dash: using down to drop and X (or Y) to jump, and using c-away to drop and up on the control stick to jump. After you dj onto the stage, just before you land, you want to hold the control stick completely horizontal and hit R (so if you're on the left ledge, you'd hold right). This way your ledge dash will be just as long as a perfect waveland.
Personally, I use down and X and have no problems, but I believe that my method leaves less room for error in your timing when compared to using the c-stick to drop from the ledge (simply because you will never fastfall using the c-stick). Thus, many players (Thomas Tipman included) prefer to use the c-stick to drop from the ledge.
Regardless of what method you choose, try to get used to DI'ing toward the stage as soon as you jump, just like when you ledgehop onto the stage. By DI'ing onto the stage during your double jump, you can get more distance out of your waveland and be able to angle your waveland for less distance when desired (very situational), as well as develop a good habit that is necessary for doing a reverse ledge dash (RLD).
For a RLD, you have to drop from the ledge, jump immediately, and as soon as you jump (or before, if you're using X or Y to jump) DI toward the stage for a split second, then snap back away from the stage and hit R. From a RLD you can bair/uair/dair (and still make it back), dj fair/dair/waveland onto the stage, dj bair/uair/dair offstage (when you are confident that you will hit your opponent) and up-b back, or just regrab the ledge as a mindgame and go from there. RLD's require some speed and good timing, it'll definitely take some practice.
