They say that I should calculate and give my zakat annually. What is the evidence of this idea? Is it not possible to give it monthly? I think it is more reasonable that way. They also say that I should give it in the month of Ramadan before Eid. Can you please provide if there is any evidence of these claims.
There is no specific time to pay the zakat. You should determine your own period.
The annual paying tradition comes from the earlier times in which most of the people were earning their subsistence annually. Now we are living the times in which most of the people are earning their income monthly. So it is much more rational to give the zakat monthly. But anyway, it is up to you and your own conditions.
Zakat has nothing in particular to do with Ramadan.