Medical expenses incurred one year, invoiced and paid the next

For a cash basis taxpayer (almost all individuals are cash basis), you are correct.

What about paying in advance. Say you pay in 2010 for a surgery to happen in 2011? Is the deduction for 2010 still allowed? If you pay for more than one year in advance, then is it still allowed?

There is a very old tax court case (1950s) that I believe is still valid. Prepaid medical expense is taken as a deduction in the year one receives treatment.

With the caveat that if it is charged to a credit card in 2010, you are considered to have made the payment in 2010, even if you don't pay that credit card bill until 2011.

The same is true for an accrual-basis taxpayer as IRC 213(a) says only paid, not "paid or incurred."