BluRay burner with a MacBook Air

One place that sells Blu-Ray burners is Other World Computing.

Here's a wall-powered drive. It's similar to the CD/DVD one I bought from them a while ago, except that it also supports Blu-Ray. The included cable is a USB-A to USB 3.0 B one and is only 24" long … so you may want to replace it with a longer USB-A to USB 3.0 B or USB-C to USB 3.0 B cable (either are easy to find on Amazon).

Other World Computing – OWC Mercury Pro Super-Multi Blu-ray/DVD/CD Burner/Reader External Optical Drive with M-DISC & BDXL Support

One application that can do Blu-Ray authoring is Toast 20 Pro.

Roxio – Toast Family

yes, but you will also need a third party app for reading and writing to the Blu-ray Discs.

If you want to make a home movie that will play in a regular Blu-Ray player, you will need a Blu-Ray authoring application that knows how to encode the video and menus, as well as how to burn the discs.