As seen in this article (though not primarily about linguistics), it promptly states:
Some neuroscientists are not so sure. They think that giving up handwriting will affect how future generations learn to read. “Drawing each letter by hand substantially improves subsequent recognition,” Gentaz explains.
Drawing each letter by hand improves our grasp ...
The Goldlist method is based on this assumption. I haven't found any peer reviewed articles on it, though.
This article (citing research by Mueller and Oppenheimer) suggests that we do learn better by physically taking notes. Take Notes by Hand for Better Long-Term Comprehension.
It isn't specifically for language learning, though.
I think I understand where you're coming from, however, I don't think you need to worry.
You already seem to have a fairly good understanding of English, and translating from Persian to English is definitely not going to harm your English skills, rather, it should help your mind better understand the difference between the two languages, and (this idea ...
I am convinced that this is true, but I don't know why. Why?
I would say memory is a very important factor here. People may have better or worse memory, but in general it's very easy to forget what you learnt a week ago, if you didn't repeat it in the meantime.
It may and probably does apply to other fields of study than language learning, but to ...
Even native speakers correct themselves when speaking.
If you say (in English or French for that matter), "when I study x" you are not saying "when I studied x". So, if your sentence is supposed to be in the past, you should make the correction if you want to say what you intended to say.
The same point is true in French for: J'ai vécu ici pour trois, ... ...
Without any claim to completeness, here are some subjective reasons why I think one should strive to correct their mistakes, once noticed:
As a language learner, you make more mistakes than you notice; they slip unnoticed and thus stay not corrected. The mistakes have cumulative effect, rendering what you say incomprehensible. Correcting a mistake, instead, ...