Can Resdient Alien buy a house in US?

Hello,

My parents have lived in the states for over a year and they decided to stay here permanently. Their green card is still pending,but EAD has been approved.

Can they now buy a house in the US for them to live in?
Are they (over 65 years old) qualified for senior property tax exemptions?
Can I be the co-owner so that in the end we don't have to pay for the heritage tax? I currently own a house.

thanks in advance.