How can I get files from my black screen iMac using a Mac mini?

Re: "I have so many photos of my children and granddaughter and thousands of songs on iTunes as well as some work related and personal records, including genealogy."

If data is important to you, you need to back it up before catastrophe strikes. There is a saying that all drives fail: it's not a matter of IF they will fail, but of WHEN. With WHEN often being long, long before 17 or 18 years pass.

There is one thing you could try. (No guarantee that it will work if the 2007 iMac is dead or half-dead.)

If you have another Mac that is old enough that it can run something earlier than Big Sur, you could try using Target Disk Mode. (2007 iMacs don't have Thunderbolt ports – so the other Mac can't be running something new enough that you would be forced to use a Thunderbolt cable.)

Transfer files between two Mac computers using target disk mode - Apple Support