Can I connect my Mid 2010 iMac 27" as a monitor for my shiny new Mac mini?

Apple says no but see:

https://discussions.apple.com/thread/251704486

It has a 2020 Mac using a 27" 2009 iMac as a display. There are posts in that thread about failures, but they were using newer iMacs with Thunderbolt, not 2009 or 2010 iMacs with Mini DisplayPort.

When a 2009 or 2010 27" iMac is in Target Display Mode it acts just as a Mini DisplayPort monitor, and will work with any computer that supports Mini DisplayPort, including Windows PCs.

Newer (2011 to 2014 using Thunderbolt) iMacs in Target Display Mode apparently do not appear exactly as Thunderbolt displays and the other Mac has to do something special to use them as displays. 2020 Macs can't do this, but Apple's article on Target Display Mode doesn't distinguish between Thunderbolt and Mini DisplayPort when it say the source Mac can't be newer than 2019. The article is marked as archived, so probably will never be updated since the error is applicable only for computers that are over ten years old.

You will need a USB-C to Mini DisplayPort cable, such as:

https://www.apple.com/shop/product/HN8B2ZM/A/mophie-usb-c-cable-with-mini-displayport-connector

No

https://support.apple.com/en-us/HT204592