[UPDATED: Here's how: you press the start button twice without your foot on the brakes. 1st press car goes into accessory mode - radio will work - 2nd press car goes into run mode. Windows, Air Conditioning, etc. Thank you Intrepid1 for the tip.]

I hope I'm not the only owner of a HCH III in this forum
Yes, it's been discontinued, but it's a great car!

A small annoyance on the 2015 (and probably 2014) which has the push-button ignition... is there a way to open and close the windows without turning on the engine?

Pressing the button once will turn the car on on Accessories Mode. Radio works, but windows don't.
On traditional cars, you can turn the key to On, but the engine is still off.

I also noticed that when you turn off the car, the windows work until you open the door to step out.

 

I think you nailed it.

Good news - if you are hanging out in the car for a while, the windows and radio will play until you leave the vehicle (unless its a very extended time)