Do I have to put the letter that I wrote to imam madhi in a special box or leave it bare with no protection cover? Can i throw it to the river without putting protection cover on it? I like writing  in a piece of paper so can i throw it like that ?

And also can I keep the letters with me instead of throwing them in the river. If I keep the letters with me , will Imam Mahdi read them ?

And what are the benefits of writing letter to Imam Mahdi? and also what are the benefits of throwing it in the river ? Will it reach to our Imam faster?

You can put it in a box or just put the letter without a box. We have narrations that the Imams encourage people to write them letters. This is to strengthen the ties and relationship with the Imam. Even if you do not write an Ariza, the Imam knows what is in your heart because God informs him.