Do normal payments from a non-qualified annuity qualify as some part or all of the Required Minimum Distributions?

No. Nonqualified annuities (those held outside a retirement account) generally have no requirement to withdraw your funds at any age unless required by the annuity contract itself. Some contracts force distributions or annuitization to begin at a certain age, generally from age 85 to 100. A few contracts do not require distribution of the proceeds until death. 

Can an account owner just take a RMD from one account instead of separately from each account?

An IRA owner must calculate the RMD separately for each IRA that he or she owns, but can withdraw the total amount from one or more of the IRAs. Similarly, a 403(b) contract owner must calculate the RMD separately for each 403(b) contract that he or she owns, but can take the total amount from one or more of the 403(b) contracts.

However, RMDs required from other types of retirement plans, such as 401(k) and 457(b) plans have to be taken separately from each of those plan accounts.

RMD