"Is there ANY way to put non-qualified (from an annuity or otherwise) inherited money into my own IRA/Roth IRA?"
You can use any cash that you receive from the non-qualified annuity just like any other cash. Cash is fungible, meaning that the cash from the non-qualified annuity is no different than any other cash you have in your pocket. Because a distribution from the non-qualified annuity is not eligible for rollover, there is no applicable deadline to do anything with the cash. It's just cash.
If you meet the requirements for making an IRA contribution (have compensation from employment, including net profit from self-employment) and, for a Roth IRA contribution, you are not limited by your MAGI relative to your filing status, you can make a regular traditional or Roth IRA contribution.
