Is it possible to add an Apple compatible remote control to an audio cable for headphones? I'm wondering whether there is some TRRS extension cable available which has a remote control for volume and play/pause that would allow to extend any headphones by a remote control, or alternatively whether I could build one myself using the remote control from broken Apple earbuds.
Yes, there are lots of them, check eBay, amazon, probably aliexpress.

Keywords I used are: iphone extension cable remote.

Most of them have a microphone as well.