Would a nursing home come back and try and sell Mom's stuff in the house to pay before Medicare takes over?

The DPOA document allows them to make the decisions & be in charge.

Now the NH is not going to go into your mom's house and do a tag-sale. The NH is not going to sell of your mom's house. But the NH fully expects to be paid for any & all services they provide to your mom. How the DPOA pays for the care whether by doing a spend-down of your mom's savings and other assets or by selling the house and paying for care that way; or applying for Medicaid if mom can qualify for it, is up to the DPOA to do and take the actions necessary for this.

If the DPOA is saying to you that you cannot go and remove things of value from your mom's house, you have to abide by their decision. Whether you think it is fair or not, they were chosen as DPOA so you have to let them make the decisions. The DPOA may need those things in place for the house to be sold or left in the home if mom does a "right of return" statement for Medicaid.

Now if you think the DPOA is not doing their job properly, you can go to court to file for guardianship hearing for your mom. But you have to be prepared to spend quite a lot of your money to do this and you & anyone in your household better have a spotless background to be able to successfully replace your mom's chosen DPOA with a guardianship.

Not necessarily. As long as she is alive and intends to return home, the MEDICAID people will leave the house alone. The POA is correct, ALL assets must be accounted for, the money from the sale offsets the MEDICAID.
Medicare and Medicaid are two different things. Medicare does not cover long term care. Medicaid covers nursing homes AFTER your own money runs out. They look back five years to make sure nothing is given away that is of value.