Can a conservator purchase the house and property from the conservatee?

A conservator has much more control over what may and what may not be done. However, in many, perhaps in MOST states a court order is required for sale of real property. And there would absolutely have to be fair market value paid, with assets moved into the conservatee's accounts quite carefully. A conservator is legally liable, and any "self-enrichment" could be prosecuted under the law. This is a question that should be answered by a Trust and Estate Attorney in your own state, not by a forum.

As long as Fair Market Value is paid there should be no problems.
Make sure the sale is done legally and could withstand scrutiny. Should Medicaid look into the sale...if any other family member decided that this was not done properly if another lawyer looked into it make sure all your "i's" are dotted and "t's" crossed.

Yes to paying FMV.

I think the circumstances would also be factored in.   Is the conservatee in a facility, with adequate finances, or on Medicaid?   Still, as ISRR states, the court would need to be involved unless there are provisions in the conservatorship documents specifically allowing this.

I believe they would have to get the courts approval and pay fair market value.