Question

1. May I help you to some more vegetables?

2. May I give you some more vegetables?
(Are both the same in meaning? Do you use #1 frequently?)

3. Help yourself to the cake.
4. Give yourself the cake.
(What about this pair? Does #4 mean #3?)

Answer 1

1 and 2 mean the same thing, yes.
I, myself, rarely use either expression. I would say, "Would you like more vegetables?"

3 is a common expression, yes, but 4 is not. I would say, "Have some cake!" or "Would you like some cake?"

Answer 2

I might also say, "Would you like a piece of cake?" To say, "Help yourself to the cake" might imply the whole cake. :)