Lease status - The old lease expired. It is no longer valid. Tenant and LL have an oral agreement for month-2-month lease. The parties have operated under this oral agreement, so it is not subject to attack by the statute of frauds, which requires leases to be in writing. To terminate the month-2-month lease, LL needs to give 30 days notice. This is a commercial lease (not resi), so the LL is not required to give a longer notice period beyond 30 days. As long that LL does not give notice, and you pay on the 1st of every month, the space is yours.
Old checks - If there are old checks outstanding, perhaps you should cancel those checks, or cancel the bank account and open a new one. I wouldn't be able to sleep at night if I were Tenant.
"Buying the Business" - I'm sure the restaurant is worth something. Goodwill, name recognition, restaurant supplies, signage, etc., all comes into play. Yes, you should negotiate some price for all of this. Except that, no lease will be included in the sale since the oral lease will terminate, and the new tenant will enter into a new one with LL.
LL eviction - In my opinion, there is no value in the LL evicting Tenant or re-entering at moment's notice. Why would they take that risk? It is very disfavored in the eyes of courts. Also, they don't want to deal with any of the restaurant equipment or hassles. It's not their business. The only thing they want is a steady transition into a new tenant.