A check payable to an individual can be deposited to that individual's business (LLC, corporation, partnership, etc.) account if the check is first indorsed by the individual and then indorsed by the business. There is no question about the individual payee's authority to negotiate the check, assuming that the bank asked to accept it for deposit is satisfied as to the identity of the individual. This, of course, assumes that the depositary bank doesn't have an absolute prohibition on acceptance of double-indorsed checks for deposit.
As to your question about checks payable to a recently purchased business, there are two solutions. First, if the acquired business's deposit account remains open, the checks could be deposited there and a check drawn on that account could be made payable to the acquiring business. If that's not feasible, the acquiring business (Never Hungry Cafe, LLC, in your example) could provide documentation of its purchase of the acquired business (Mary's Cafe), indorse the checks as drawn and as Never Hungry Cafe, LLC, and deposit them. The management of the business should take steps to have any checks made payable to the new business name in the future.
If Never Hungry Cafe LLC wishes to continue operating the acquired business as Mary's Cafe, it can file and provide copies of any necessary paperwork to operate under a fictitious name in your state (a "DBA" filing), then deposit checks payable to Mary's Cafe to the Never Hungry account or into a separate account for the Mary's Cafe operation, whichever your bank and the LLC agree to.