I have done this quite a few times, so I'll let you know my experiences.

yes, officially, you have to let the bill cycle cycle. but they have made exceptions for me. I paid off online the same day I made a charge, so I do not have a balance. I then called customer service (and his supervisor), and was turned down. they insisted I had to wait till the bill comes. I called back again a day later, and a supervisor made an exception for me to use the rebates before the billing cycle.

I am suprised you have a $35,000 credit line on a new GM card!! hope this helps.