Can I record a dvd off my computer and play the sound track in my cd player?

You "rip" the DVD. There are tons of programs that do this for free. I use Handbrake, but by default, it doesn't rip copy-protected DVDs (which are pretty much all DVDs you buy from a store). There is a workaround, but you'll need to look that up on your own.

Once you have the "ripped" file, you can then use any program that supports audio/video manipulation software to extract the audio track.

No.

A CD player cannot play any DVD.

However, any DVD player can play CDs ---

that is a requirement of licensing agreements.

Likewise, any BD player can play both DVDs and CDs.

You will need to make an "audio CD",

and even that might not play if the player is to old.