From what I understand, in order for a customer to be part of more than on pricing plan they must have a separate customer_code for each pricing plan. This seems a little odd to me as I would think customers would want to buy more than one subscription product on the site. How do you recommend best supporting multiple subscription products for the same customer?

At the present time, this is correct. Without understanding your business better, I'm afraid I can't get very specific but it may be possible to used tracked items in a creative way. Is this a typical scenario for your system? If not, maybe a workaround is just the thing for this edge case.

Will CG support different customer_codes with the same customer email?

If this doesn't work now, it should.

I'm assuming that since it's a separate customer code it will show up as a separte billed item on their CC.

This is correct.