Translation from a french book by Serge Brussolo

Underground storm

My father was a criminal on the lam: [I had to look this one up, I've never heard it before as it's American slang - "on the run" would be a better translation as it's universal for all us anglophones!] at least, this is how my mother always put it. On evenings when she would feel to behave as a mother ["when she would feel like being a mother" might be a better translation, it implies that she doesn't always act like a mother], she would tell me, whispering, that daddy had been part of the Weather Underground. He was one of these weather men who, a long time ago, had made the US government shake ["tremble" would be the word we would usually use, they mean the same it just sounds more correct :)] by advocating for the civil war. I was 6 years old. This Weather underground would move my childish imagination and make me ["and I would"] picture underground storms, hurricanes devastating the sewers of a city and making the buildings 
collapse.

My father, whose name I am ignorant of ["I don't know" or "I've never known" if you're feeling a bit more poetic.], had fled the USA two [you can leave out the 'two' if you want, we usually just say 'seconds before'.] seconds before the FBI would catch him [would have caught him]. From this time on [that point onwards], he had gotten lost into [he lost himself in] the wilderness, the icy deserts, where no federal agent would have the guts to come ['to look' might be a better sounding  word.] for him. His physical condition and a particular talent helped him in this ['in this' is a little vague, might be better to refer back to what it is he's doing - 'in his life of solitude' might sound better]: he was an outstanding climber, a matchless mountaineer ["without peers" sounds better]. In order to survive, he became a high mountains guide and left for the other side of the world to practice his profession. He was paid quite a price [if you want to use 'quite' the phrase 'quite the sum' is the common expression when talking about money]  to train Japanese businessmen and take them to summits such as the Chimborazo, the Aconcagua, the Kibo, the Godwin Austen or the Nanda Devi.

My mother, Anne Katz, met him during an excursion. She was French but lived in Switzerland. Lately graduated from the Beaux-arts [I don't understand what's going on here, did she graduate late from the Beaux-arts or did she just graduate recently?], she wrote tales for children that she illustrated herself. She was a marvellous drawer but she lived out of touch with the real world, in a fantasy filled with dwarves, fairies, unicorns and other fancies which delighted me as a child. She had painted on the walls of her office a fresco representing a medieval castles landscape with small misted valleys where legions of gnomes fought in an unpredictable battle. Quite strangely, this fantasy world seemed more real to her than the one in which she lived. I saw her crying when she had to make one of her characters die whereas she paid little attention to the accidents I suffered (a broken leg, a peritonitis, a concussion, etc…).

It was quite strange for a little girl around ten years old to compete with individuals existing only on paper. I would often sneak up in her workshop to observe my enemies, whose faces taunted me from the center of the sheets tacked on the drawing table.

Your level of english is excellent, any of the parts that aren't crossed out are just rephrasing sentences to make it sound like an anglophone wrote it. Overall, you're pretty solid there was just one instance where the past tense threw you a little bit. Keep it up, hope the correction is of some use!