First and foremost, since you have been served with a 30 day notice to quit, you should consult with an attorney ASAP! and then sort out title to the house and distributions pursuant to the will and trust.
The name is not an issue if its clear who was intended. That said, if the house is owned jointly, each has an equal right to occupy the house. The only way to sell would be a partition action if the parties dont agree. In the end, the sale can be forced. You really need to consult with an attorney.
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Your question points out the flaws in any co ownership of real estate. Any co owner can force a sale through partition. Each co owner has a right to possession and to occupy the residence. You need to engage an attorney to stop the UD action, and to explain these concepts. Do not wait. Hire counsel to help you now.
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