To (try to) answer the questions (spoilers ahoy):
1: The Princess of the Chrystal was one part of the soul of Momoka, who was trying to stop that pink-haired guy (forgot his name). Hence, as the Penguindrum was also an item that could change fate (though in a lesser way compared to the diary) and she wanted to use it to for that exact purpose (it can be assumed she had a feeling she knew what was gonna happen, i.e. Kanba's fall), she wanted the two brothers to go fetch it before it was too late so she could stop the doc's plans by having the Penguindrum returned from their little family (which shouldn't have existed in the first place).
2: A penguindrum (a.k.a. the apple) is an instrument to change fate. In this case, it saved two people who would have otherwise died and thus changed the way the world was supposed to go. A way that paved the way for the plans of the big bad.
3: The penguins are helpers of Momoka, much like the bunnies were for her enemy.
4: Mario-san held the other half of Momoka's soul, tasking Masako with retrieving it as well (because due to her relation with Kanba she was already entangled in this in the first place, fate and all that).
This is all pure conjecture on my part, mind you. But hey, part of the fun of the show was figuring out what the hell was going on. It was like a giant puzzle dumped in your lap and then being told you could go figure it out yourself how it all fit together. Personally, the show was one of my favourites last year (just behind Steins;Gate and Madoka, really) and I loved all the weirdness. Because hey, it did give me something I don't see every day.