can my MacBook Air connect directly to a DVD player

I've had a DVD burner/player connected to my iMac or Macbook Pro whenever I want to burn a CD or DVD. I can also play them using it. There will be no icon on your desktop for it unless you insert a CD or DVD - it will show an icon for the media. Mine is an older version of these - just make sure you get one that has the connection that will be compatible with the one on your computer.

https://eshop.macsales.com/shop/firewire/optical-drives

If you want to copy a DVD, you need software to do that: Roxio Toast is excellent for that.

Maybe. Depends on the DVD player, and what you are trying to accomplish by doing that.

You can connect your Mac to an external DVD drive and play DVDs on your Mac using the DVD Player app: link ➜ https://www.amazon.com/Best-Sellers-External-CD-DVD-Drives/zgbs/pc/1292121011

You cannot connect your Mac to a set top DVD player like the type you connect to a TV, and expect to see its image on the Mac without additional hardware.

In that case you need a DVD / CD writer / burner drive to read the DVD and make a copy using 3rd party software. A set top DVD player will not let you make a copy of the DVD on the Mac at all.