If I place into Calculus 3 (Multivariable Calc) at my college, would medical schools be okay with me skipping Calc 1 & 2 and just taking Calc 3 and a stats class?
You should be fine with Calc 3 and stats.
Agree, a more advanced calc class and stats is fine. Maybe also try to get the credit for entry level calc put on your transcript (credit by exam or some such), it would look good.
As far as I know, only Harvard HST MD track requires multivariate calculus/linear algebra as a prerequisite for their MD track. Harvard new pathway MD track only requires Calc 1 and stats.