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?)

1 answer

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?