Mom is the wife of a living Korean war veteran, they are currently living apart, but not legally separated. Is she entitled to VA benefits?

Where I live, there is a local veterans' benefits office at the county courthouse. Their services are free, and they interface with the national Veterans Administration. Here, you call the number and they make a phone appointment to discuss what you need to do. Have dad's DD-214 form handy; it gives information about his service and discharge. If you don't have it, perhaps the rep can get it. The rep will come to the home with forms that are necessary to sign and will then submit them to the VA for you.

You can also call the VA in another way at a national number, but I prefer dealing with them through the local office because the rep takes a more personal interest than a faceless person on the phone.

If dad gets a military pension, find out if mom is an annuitant so that she can collect a portion of his pension when he passes. He would have had to sign her up for that and pay for it over the years.

I don't know if she qualifies for home care. She (and he) may both be entitled to burial in a national cemetery.

Good luck in finding out what you need to know.

As long as she is a WIFE she has all the benefits due a WIFE.
If VA benefits supply home care to a wife, then she should be able to receive some home care, in my own humble opinion, but I would check with a VA advisor or social worker. As you might know, unless we face this exact situation, our opinion is only that--opinion.
I sure wish you the best.