Do USB-C to Thunderbolt 2 cables exist?
I don't think a straight-up cable exists, no. However, you can get an adapter/dongle that works perfectly well for your scenario. You'll need to Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 adapter:

https://www.apple.com/us_smb_83039/shop/product/MMEL2AM/A/thunderbolt-3-usb-c-to-thunderbolt-2-adapter

You can read more about the adapter here:

https://support.apple.com/en-gb/HT207266

Note that Target Display Mode is not using the same kind of signal as mini-DisplayPort - therefore a USB-C to mini-DisplayPort cable will not work! It needs to be an actual Thunderbolt 2 source.