For medical expenses, I don't have every receipt. Will the IRS accept a ledger of payments from each provider as well as insurance and credit card statements?

You wouldn't necessarily need an individual receipt for every service; an account statement showing payment for all services during the year from a particular provider is acceptable.  As are printouts from your local pharmacy for prescription amounts paid throughout the year.

An insurance statement -- which in itself probably wouldn't prove payment, merely the amount you were "responsible" for -- coupled with a credit card statement or cancelled check, would also work. 

Should the IRS ask for supporting documentation, the key is to have documentation proving amounts actually paid, rather than invoices or bills.

As a follow-up question: if a medical procedure was done in 2015, but paid for in 2016, can I use it as a medical expense for 2016.  In other words, is the tax year for a medical expense based only on when it was actually paid?

Yes, you claim medical expenses for the year paid.