Part 2 - Please read part 1 first.
-to study a little grammar rules every day

"Rules" is plural, and you can't have " a little" of something that is plural. You would need to say this as either:

To study a little grammar every day or less formally To study a little bit of grammar every day

or, keeping it as plural:

To study a few grammar rules every day
-to learn one or two new words every day
-to listen to English audio as much as possible

Again I'd probably be a bit more specific there; maybe "recordings and broadcasts"
-to write some sentences with what I learned during the day
-to try to think in English in my head... but this is the hardest think thing! :)

You think with your brain, which is in your head... so saying "Think in English in my head" is redundant. Unless you're a politician. Some of them use a different part of their body for thinking.

I'm still amazed at the progress that you've made in only three months; it's amazing how well you can write in such a small amount of time.

My study plan

My study plan

1/ "Since when" is used when you are asking about the starting time of something. For example, "Since when have we not been allowed to park here?", which is another way of asking "When did that rule start?" In this case you are giving the time period ("three months"), so we already know "when". You therefore say "Since I started {something} three months ago". 2/ "Since I started with English" is OK, but it's a little informal. "Since I started studying English" or "Since I started learning English" are more common ways of saying this, but the meaning is still clear enough. 3/ "I study" is the present tense. However in this case you are talking about what you have done in the past as well. The form that I have shown is called the "present perfect continuous" (if you're a grammar geek and want to look it up) and it is used when you have been doing something over a period of time, up to and including now. There is a similar form that people often use called the "present perfect" which is "I have studied every day". Strictly speaking it's used when the thing that you're doing will be finished at a particular time, but in this case it can mean "I have studied up to now". The grammatical terms (present, perfect, etc) aren't exceptionally important. The main thing that you need to know is that English is just a little bit more aware of past, present and future than Italian is. In Italian you can use the presente for almost anything in the near past, present or future. In English you can sometimes do that, but not always.

I study a little grammar or write on Italki at least one or two hours per day.

"I study a little grammar" is another informal expression. "I study some grammar" or "I study a little bit of grammar" are both less informal, but the form that you have used is quite common. An expression like "I ate a little Mexican last night", for example, would be clearly understood as meaning that you had a small amount of Mexican food, not that you ate a short person from Mexico. Similarly a more formal form of "I write on Italki at least 1 or 2 hours" would be "I write on Italki for at least... etc" However the informality works well in my opinion because it's an informal article that you've written.

Other than that I listen to English audios as much as possible.

You wouldn't normally see "audios" used in that way, because it's not clear what they are. If it's music you would say "songs", if they are radio or TV programs you might say "broadcasts". Your meaning would be understood by a native speaker but it wouldn't be clear.

I Always play at work some English videos in the background at work... even if I don't understand them, I try to immerse myself in this language.

1/ The problem with "at work" being where it is is that you are splitting up the verb ( "play" ) and the thing that the verb is acting on ("some English videos"). If you move "at work" to the end of the sentence, you are keeping the verb with its object.

I wake up early in the morning to study because I don't have much free time during the day. Sometimes I'm demoralized and tired but when I see my improvements, I know that I must not give up!
So what I do to study English is:
-to write on Italki

And I'm out of space again. To be continued.