Coompatibility

Yes, but the iMac will not be able to display the full resolution the Mac Mini can produce so you will have to scale down the display to the iMac's capabilities.

Easier, and better to just buy a new monitor that can support the Mac Mini's full display capabilities.

You will need a USB-C (mac mini) to mini-DisplayPort (2009 iMac 27 inch) cable to make Target Display Mode work. for 2009 and 2010 iMac 27 inch models.

I was never able to get my mid-2011 Mac mini to do target display mode with my Late 2013 27in iMac using any cable combination suggested by Apple. It would work perhaps 2 out of 10 tries, and I finally gave up on that approach.

I purchased an HDMI dongle that spoofs the Mac mini into believing an external monitor is attached to it, and then set up Screen Sharing on the mini (macOS 10.13.6) with my iMac. When I use the Mojave Screen Sharing application on my iMac, the Mac mini screen appears at at very usable size and clarity — and both Apple extended keyboard, and magic mouse 2 work just fine with the mini while connected to the iMac. Not a full screen solution as would be TDM.

In theory yes, in practicality just buy an external display. Use your iMac as a display with Target Display Mode