Powering an old (2015) imac 24 all in one from a new Mac air ?

If you're asking if you can use a 21.5" 2015 iMac as a docking station for a new MacBook Air, the answer is no.

Apple dropped Target Display Mode in Late 2014 when the first 27" 5K Retina iMac came out. No 2015 iMac has it. In addition, Apple placed restrictions on newer Macs such that none of them can use ANY iMac as a Target Display. Even for configurations where Target Display Mode is supported, it only grants access to the screen – not to other components of the iMac such as the hard drive.

You could use iCloud to synchronize some data between the 21.5" 2015 iMac and the new MacBook Air, letting you start work on one machine, and later continue it on the other. How well this would work would depend on whether you were using compatible applications on both machines. You can upgrade either of the 21.5" 2015 iMacs as far as Monterey, but no further.

“wondering if a new 13 inch Mac air (M3 with 8 gb ram 256 g hard drive) would plug in and run this old Mac ? Or would that simply mirror what the Mac air laptop is doing - or would the cpu on the laptop run the iMac 24 - Allowing me to continue using the old machine with its large screen and hard drive storage and then simply disconnect and use my Mac air as laptop on the go..”

Neither.