1) Carrot's dream appears to be to explore the world outside Zou. She's repeatedly shown fascination with the "wonderland" of the sea and expressed a desire to know and see more. Plus what with Pedro's speech at the end of the last chapter, now she has a reason to specifically do so with the Straw Hats.
2) I'll admit that's a good point, although she did have the whole Mirror World mini-arc to herself along with Chopper. It's worth pointing out that Brook did next to nothing himself during the Thriller Bark arc before he joined.
3) Again, Pedro. Happening right now.
4) More than likely she'll stow away and insist on joining the crew afterward, like Robin did.
5) Besides her electro powers, she's faster than Zoro with a unique acrobatic fighting style. Plus, her ability to jump like a mile into the air and her greatly enhanced sight mark her as a candidate to be a capable lookout, a traditional pirate role they've been sorely missing.
She's already building up strong relationships with the rest of the crew, especially Chopper, she more or less
already acts like she's part of the crew, and there was even one point in the anime where she
flat out asked to join them (although that was just filler so it's canonicity is dubious at best.).