I would use 1, 3, and 6. The others are awkward or incorrect.
4 could easily be changed for common use:
Would you help me with the dishes?
1. Would you give me a hand with the dishes?
2. Would you give me a hand to do the dishes?
3. Would you give me a hand doing the dishes?
(Are they all grammatical and do they have the same meaning?)
4. Would you give me help with the dishes?
5. Would you give me a help with the dishes?
6. Would you give me some help with the dishes?
(Does #1 mean #4 or #6? #5 is not correct, is it?)
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