The landlord can evict you; however, if you remain on the premises, continue to pay rent as a "holdover tenant," and the landlord acquiesced or does nothing to evict you, you become a tenant under a month-to-month lease.
Regardless of your status, you can inquire with the landlord into whether you can be the new tenant. You should enter into a new written lease, if possible.
If the landlord decides to evict you, the process is costly and lengthy and at the end of it, the landlord must provide you with a Notice of Right to Reclaim Abandoned Property should any personal belongings of yours remain on the premises.
You should consult legal counsel regarding the specifics of your case.
Yes, but the answer regarding how much time depends upon whether there is still an operative lease in effect (when did your father pass away?), and if so, what does it say, and whether you are still paying the rent and whether the landlord has accepted the rent.
Generally, if the landlord accepts the rent from you, it creates a new periodic tenancy (month-to-month), which would require the landlord to give you 30 or 60 days notice before being able to evict you.
The information presented here is general in nature and is not intended, nor should be construed, as legal advice. This posting does not create any attorney-client relationship with the author (who is only admitted to practice law in the State of California). For specific advice about your particular situation, consult your own attorney.