Rental contracts in the Dominican Republic usually last for one year - though rolling monthly contracts can be negotiated (a good idea if you are new to the island). You might be required to allow your landlord access to your accommodation on a monthly basis to check everything is in order - this should appear as a clause in the contract. The landlord is also required to fix anything in the apartment which has not been broken by the tenants.
It’s not unusual for landlords to demand 2+1 or 3+1 of the equivalent of the monthly cost of rent. What this means is that two to three months rent will be paid as a deposit plus an additional one month's rent as a fee towards the landlord’s lawyer. The proportion that makes up the deposit will usually be placed into a bank account until the end of the lease, when you will receive it back.