1. A kind girl, she was loved by all.
2. Because she was a kind girl, she was loved by all.
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Does #1 mean #2? Or in #1, is "A kind girl" in apposition with "she"?
3. As it was rainy, I had to stay at home.
4. It being rainy, I had to stay at home.
5. It rainy, I had to stay at home.
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Are the three sentences all grammatical? Can we omit 'being' as in #5?
1 and 2 mean about the same thing, and yes, you're right about the apposition.
Only 3 is correct. Omit 4 and 5.