Does my agreement expire after 12 months? No. The agreement continues but only for 30 day periods. After each 30 day period either you or the landlord may give appropriate notice to change the terms of the agreement going forward. If either of you disagree with the changes, you can terminate your tenancy with the proper notice. The agreement does expire in the context of your question in that the Landlord may give notice to raise the rent after the initial 12 month term.
Does the word "continue" as opposed to "renew" mean it does not expire? See answer above.
Does this allow the landlord to raise the rent at will after 12 months? Yes with proper notice.
Or is the rent fixed until a 30 day rental termination notice is given? You would not receive a rental termination notice unless the Landlord wanted you out of the property. The rent is fixed at the monthly amount as noted in your lease until proper notice is given to change that amount. The Landlord does not need to "terminate" a month-to-month lease in order to raise the rent.