It is up to the woman to initiate physical contact, including the shaking of hands. If the woman extends her hand, then the man can shake it, but the man is not supposed to initiate the process.

The older woman offers her hand to the younger one.

The woman is not required to stand.

But the woman can stand if she desires.

Totally depends upon the business etiquette that you choose your family to abide to. My answer would be:

Most proper families lead, and teach their children to make good impressions when doing business. They do this when their children are young.

Men AND Women should stand and shake hands. Men stand, women sit????? This is NOT the days of the pilgrims.

They should ALL STAND, and say their names and thank each other for the reason that they are at the dinner table. Thank the God that you believe in for giving you the gift of life.

Good luck.

You should. It shows courtesy.