Laymans guess , retired LL:
Service of notice requires proper delivery --not that party even be awake
As a LL but not in TX, if I goofed on a notice the judge made me eat it and do it over again for the simplist of errors --quickly learned not to make mistakes!
I think her LL goet it wrong--TX allows for LL to terminate by the day with 30 day notice I have trouble following if a notice to vacate by 15th and a demand to pay for full months is totally consistent with TX law as to notice--you go read TX rules, your decide.
Bottom line the executor will quickly find out how to get it right--so Mom would be wise to work it out sooner rather than later.
Sorry but a oral promise of a life estate is most unlikley to be enforceable, won't happen -- If there were any written docuements of any sort as to Mom having a long term lease-life estate in return for some specific payments which were in fact made --Mom might have a bargaining chip and claim IF she filed a claim against estate promptly. Dig about ?
As a technical aside, unless the executor has filed all the proper paperwork to be FORMALLY appointed by court as executor and has been so appointed by order--then he lacks proper standing to tell her to do anything! (Not hard to check court records)