Copying files from 2013 iMac with damage screen

If you have a standalone monitor with a DisplayPort input, you could run a Mini DisplayPort to DisplayPort cable from the iMac to the monitor. You could also get a Mini DisplayPort to HDMI adapter and run a cable to a HDMI-equipped monitor or TV.

Note:

• 2013 iMacs do not support external displays with resolutions greater than 2560x1600 pixels. (But you might get a signal on a 4K monitor or TV if it supports lower resolutions and the iMac picks one of them.)
• To get the menu bar and Dock to show up on an external dock, you will need to enable mirroring, and hope that the iMac and monitor will work at whatever compromise resolution the Mac picks. Since you can't see to set up mirroring using the GUI, you'll need to use the keyboard shortcut. I believe that it would be Command-F1.

One option that none have mentioned is the fact that you can open the damaged iMac and remove the internal drive. Put that drive into an external drive enclosure and access your data.

If you are not technically skilled yourself, any shop should be able to remove that drive for you at a minimal cost.

You can recycle the rest of the computer afterward.

If you google “Mac repair guides” or “Mac disassembly” you can find guidance for removing the drive.

Apple Silicon Macs should be able to connect to Intel-based Macs running in Target Disk Mode, but with a rather significant limitation.

The connection must be a Thunderbolt one. (Follows from the Big Sur clause, since no Apple Silicon Mac can run anything earlier.)

This is expensive to make to Macs with old-style Thunderbolt 1 & 2 ports.

You can copy the files from your 2013 to your MBA 2015 using Target Disk Mode. I don't remember if your Silicon 2025 MBA can utilize Target Disk Mode. Read the document in the link and see.

If perchance you had enabled either File Sharing or Screen Sharing on the 2013 iMac you may be able to access it from your MBAir(s) via your network one of those ways.

For more information see:

• Set up file sharing on Mac
• Share the screen of another Mac

You cannot use either MacBook as a cabled display for the iMac. They do not accept video input.

You might be able to put the iMac into Target Disk Mode and use it as a USB drive for the older MacBook Air, provided that both computers are running something earlier than Big Sur. Once either is running Big Sur or later, the connection must be a Thunderbolt one - which would be more expensive.