DP points are pretty ridiculous. Every time you move your home point, you use a DP, basically. Every action you do out of your home point moves your home point to the square you did that action from. Or in other words, doing actions out of your home point uses DP. Thus, it's better to perform actions from squares where there's one or more targets available.
Each stage has it's own bravery pool, which is increased through various means. If someone gets broken, the person who triggered the break is awarded all those points.
The right and left side of your screen have to do with you and your opponent's accessories. You know all those accessories that provide multipliers to your currently equipped accessories? Those symbols indicate which are active, with the number indicating the overall multiplier currently in effect.
For example, say your opponent has a "Defense + 10%" accessory, then an HP = 100% 1.5x, HP > 80% 1.2x, HP > 50% 1.2x, and a Near Death 2.0x accessory. When the battle starts, he'll have four symbols with three of them filled in. That means the three HP > X% accessories are active, and the multiplier would be at 2.2x. As you decrease their HP below those levels, the symbols will fade and their multiplier will steadily go down, until you get them into red flashing HP, at which point they'll jump to a 2.0x multiplier. Taking a look at how your opponent's multipliers work can save you a lot of trouble. Those that have a bunch of HP > X% multipliers should imply that you should hit with HP attacks regularly to continually weaken their offensive capabilities, while some have a ton of HP < Y% and Near Death multipliers, which might mean you want to gather a substantial supply of brave before going in for a single killing blow.