Let's take it step by step. Disclaimer: I'm neither an accountant nor a lawyer. You need the assistance of both.
First, as executor, your sister has a fiduciary duty to the estate. And she has a responsibility to execute your mom's directives. What your mom requested is pretty standard, and a pretty good way to handle things. I personally would suggest you contact the lawyer who helped your mom create the living trust and get his advice on how best to make sure that your mom's wishes and directives are followed.
Had one of your siblings wished to buy the home--or if one still does--that probably can be accomplished within the terms of the trust. It's appraised, per her wishes. It's put on the market, per her wishes. Then one of the siblings can buy it for full, fair market value. That'd be entirely in accordance with her wishes, and would be fair to all the children. The problem arises because the executor is trying to cut a side deal with another sibling.
Again, start with the lawyer who drew up the papers. If he declines, ask for a referral. Or look for another good estate planning lawyer in your area.
As for the loan company wishing to see a copy of the trust: When property ownership is transferred, it routinely would trigger the "due on sale" clause in the mortgage. There are exceptions under the Garn St. Germain Act--and transfer into a trust, and certain other related actions are permitted. But the mortgage requires the lender's pre-approval before a transfer takes place. They've got a vested interest in what's going on. They want to make sure that ownership isn't being transferred in a way that would trigger the due-on-sale clause that isn't protected by one of the exceptions in Garn St. Germain. After all, the security for their loan is the property itself. Check with a lawyer, but I suspect the loan company has every right to view the trust document.
So: Check with the lawyer who did the living trust for your mom. Find out how your mom's wishes, as reflected in the trust, can be carried out. And ask whether the lender can see the trust document.
Hope that helps.
Does the lender have a right to ask to see the trust? Of course they do. The lender wants to know who the trustee is and the powers of the trustee wields. The lender needs to know they are dealing with the person who is legally able to make decisions.