First, I haved travelled 7 hours with only one break to eat, and endured an hour of traffic jam to come late to the inventory of my new apartment.

had = past tense of have; Traffic without the jam flows better here.

Just after my quick inventory, I met my new flatmates, and one say immediately said " doid they saytell you about the cockroaches, bedbugs, and water leaks ?

said = past tense of say; sentence flows better with immediately in front of said; did = past of do; tell is correct here instead of say; the in front of cockroaches, bedbugs, and water leaks

", it's a nightmare !

I didn't eat dinner/ dine after all this adventures, and stayed in my bedroom, almost all the time in my bedroom, for three days.

dinner is not a verb; adventure is singular here; almost all the time after

In fact Lyon is not so dramatical, with a lot of courage and a little bit of beautiful weather, iI have visitsearched Lyon to find a better neighbourhood to live sinceafter today.

dramatic instead of dramatical; I is always capitalized; if I am understanding correct you were already in Lyon meaning you cannot visit it.

I came here to study, I just begian a new coursuse so in summary that's like a totally new departure, like I started from 0 with nobody/ without noanybody to reassure me.

began is past tense, course is singular and modern version of cursus; thats needs ' to imply that it is short for "that is";

I arrived a week ago in Lyon, and it was a difficult week to discover a new city.

Your text is correct, but a more natural order for the phrase would be, "I arrived in Lyon a week ago..."

First, I have travelled 7 hours with only one break to eat, and endured an hour of traffic jam to come late to the inventory ofto arrive late to my new apartment.

"Traffic jam" is quantitative while "traffic" is not. So you could say, "there was traffic" or you could say "there was a traffic jam." Another small nuance, traffic implies slowness but a traffic jam is a little stronger, meaning traffic could be completely stopped or barely moving at all.

Inventory = inventaire, like a stockage of items. Not sure what you mean here, but you can simply say "my new apartment"

Just after my quick (inventory?), I met my new flatmates, and one say immediately " doasked, "Did they saytell you about the cockroaches, bedbugs, and water leaks ?

Again not sure about what you mean with inventory, but from the context, maybe you are talking about moving in (bringing your stuff inside the apartment to live inside). If so, you might say, "Just after quickly moving in my stuff, ..."

Since your flatmate was asking a question, you'd use "asked"

There are no spaces after question marks or exclamation points in English

", a A nightmare !

Here you finished a sentence (the quotation, finished by the question mark) and are starting a new sentence, so A is capitalized.

I didn't eat dinner after all this adventures, and stayed almost all ththe entire time in my bedroom for three days.

Your process of traveling and moving in could be thought of as one singular adventure

In fact, Lyon is not so dramatical, w. With a lot of courage and a little bit of beautiful weather, i havetoday I visited Lyon to find a better neighbourhood to live since today.

I came here to study,; I just begian a new cursus sriculum. So in summary that, it's like a totally new departure, like I start from 0 without nobody to reassure me.

I arrived a week ago inLast week i arrived to Lyon, and it was a difficult week to discover a new city.