It's just her nature to use her looks in an attempt to get what she wants, plus it could be to annoy Amy due to her liking Sonic so much.
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Yes but she's only a replica of the real rouge like in the story.
Rouge appears as a boss at figure skating. After you beat her, you can buy the rouge costume in the shop for your mii.
Shadow the Hedgehog is the equivalent of sonic. Rouge the bat is the equivalent of tails. and Omega is the equivalent of knuckles the echidna.
He is called Silver the Hedgehog. He is silver not white but Rouge is also partially white. If you mean the girl with bat wings, that's Rouge.