WDo you want to know what I see accrossthrough my window.?

I see what you're going for, but “what I see through my window” is not a complete sentence. I phrased it as a question but there are a million other ways to write this; the most important part is that the sentence is complete.

I am currently at work, and I have a big window ton my right.

“On my right” isn't horrible but “to” is much more natural

I can see that the sky is cloudy and that it is rainning.

Close to the window, tThere are three trees near the window.

Near the window - prepositional phrase. Putting a prepositional p Gee ease at the beginning of the sentence like you did initially puts the focus on the preposition and pulls it away from the rest of the sentence.
Ex.
After the war, people were kinder to each other. - emphasizes the change, this kindness did not exist until the war ended
People were kinder to each other after the war. - no specific emphasis but the focus is on the kindness and community brought about by the war

They have green leaves, meaning that summer is comming.

Next, there is a busy road, where a lot of cars and trucks roll on this road every minutesdrive by.

A little awkward

These trains are usedPeople use the trains to commute to Paris toand the suburbs.

“Are used” is the passive voice which you should only use where necessary, especially unto you get a really good understanding of active and passive voice. It's tricky even for native English speakers so I wouldn't mess with it too much for now.

I am currently at work and I havethere's a big window ton my right.

You can also indicate the physical location "I'm currently at the office/at my desk at work and there's a big window to my right"

I can see that the sky is cloudy and that it is rainning.

Close to the window, there are three trees.

I think this is okay, you can also say "Near the window, there are three trees"

They have green leaves, meaning that summer is comming.

Next, there is a busy road, a lot of cars and trucks roll on this road everyby every few/3/5 minutes.

You can indicate the number of minutes if you want to be more spcific, or just say "every few minutes".

I don't like thait because ithey makes a lot of noise.

These trains are used to commute to Paris to the suburb.

What I see accross mythrough the window.

à travers = across, only when you're talking about crossing an open space, like a field, a street, etc. Otherwise, à travers = through.

I can see that the sky is cloudy and that it is rainning.

They have green leaves, which meanings that summer is comming.

The way you wrote it works, but this sounds a bit better stylistically. We tend to avoid multiple occurrences of "-ing" in the same clause.

Next to it, there is a busy road, and a lot of cars and trucks rolldrive on this road every minutes.

"next" means adjacent in time, you have to use "next to [something]" for adjacent in space
"rouler" ne se traduit pas directement en anglais, c'est toujours "drive"

These trains are used to commute to Paris tobetween Paris and the suburbs.