iMac (21.5-inch, Mid 2014) does my iMac supports target display mode ? i want use this iMac as secondary display

• 27-inch iMac models introduced in 2009 and 2010: Use a Mini DisplayPort cable to connect your iMac to a Mini DisplayPort or Thunderbolt or Thunderbolt 2 port on the other Mac.
• All non-Retina iMac models introduced in 2011, 2012, 2013, and mid 2014: Use a Thunderbolt cable to connect your iMac to a Thunderbolt or Thunderbolt 2 port on the other Mac. If the other Mac uses Thunderbolt 3, you can use the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter with your cable.

Target display mode requires macOS High Sierra or earlier on the iMac used as the external display. It does not work with macOS Mojave or later.

IMHO that would be an extremely foolish move but if you have a Time Machine backup that has 10.13.6 or earlier you could restore to that. If I were you, just buy an external display (they're cheap) and get on with it.

What version of Mac OS is installed, if it's 10.15.x as your profile indicates then it's not going to work, you can only use 10.13.6 or earlier.

Yes.