Looking for creative, effective and economical ways to upgrade my macbook air to the present day M1 capabilities.

You cannot use the MBAir as the keyboard & monitor for a Mac Mini (or any other Mac, for that matter).

A mid-range Mac Mini (16GB RAM, 1TB SSD) with an Apple keyboard & mouse comes to about $1400. But you would also need a monitor - figure $300-$600 for a non-Apple monitor. Total $1700-$2000.

A similarly configured mid-range iMac (16GB RAM, 1TB SSD) with the included monitor, keyboard & mouse comes to about $2100.

A similarly configured, refurbished M2 MBAir (13.3" or 15") is about $1600.

You would not necessarily need an external SSD if you configure any of the above with a 1TB internal SSD.

No not possible because the new Mac mini can not work in conjunction with the 2015 MacBook Air or startup and run macOS Catalina.

What you need to do, is move from the 2015 MacBook Air to a new Mac mini or to a new MacBook Air.

see > Transfer to a new Mac with Migration Assistant - Apple Support

What are your requirements and goals here?

A Mac mini has a different target than does a MacBook Air. Bigger and with more connectivity and a lower price and not portable, as compared with smaller and connection-constrained and more expensive and far more portable, to start.

Mac mini M1 is way faster (2390 / 8727) than MacBook Air 2015 (i5: 867 / 1590, i7: 1071 / 2020), too.