Yes, you can, just like any other catchable Pokemon.
Though the catch rate is reduced, it is completely possible to catch Dialga without lower its health. It is recommended to use a Quick Ball on first turn to improve your chances. Of course, the Master Ball can be used to catch Dialga at full health too.
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20%.
neither of them is better, they are both the same, but in diamond, you catch dialga, and in pearl, you catch palkia, and you can catch different Pokemon in each game for example: you can catch this Pokemon in diamond, but not pearl but: you can catch this different Pokemon in pearl, but not diamond
You catch dialga/palkia on spear pillar inside of Mt. Coronet for diamond or pearl
You can catch Palkia in Pearl and tranfer Dialga to pearl from diamond.
Easy if u dont catch DIALGA and you kill it, it will say dialga vanish or soming