What is the difference between Оставшиеся в живых and Выживший. Why wouldn't you use the initial one?
The same thing. The first one is just too pompous and doesn't fit in the flow of your story.

What is the the difference between вдруг,внезапно and неожиданно?
внезапно - suddenly, abruptly
неожиданно - unexpectedly
вдруг - suddenly but it is colloquial so I didn't include it in your text, which is definetly in official style, cuz it reminds me of a news report

Один мость в юга-западном Китае рухнул vs Рухнул мост на юго-западе Китая >> Why is it wrong to use the adjective юге-западный?
В юго-западном Китае рухнул мост - this is okay but it sounds like юго-западный Китай is like the brand new country or a special officially separated or divided area of China
На юго-западе Китая рухнул мост - this doesn't claim anything besides the fact that the bridge collapsed somewhere in the south-west of China.

машины начало трясти vs машины началися вытрясти >> Here why you chose to use начало instead of началися?And why do you use the neutral form of it?
Firstly, началися is not a word, начались is
Secondly, машины начало трясти - начало doesn't belong to машины but to some unknown dummy subject, which was apparently the source of the trembling of the cars. Literally it corresponds to this English sentence - Something started to shake the cars.
Thirdly, I could have named the cars to be the pefrormers of the "starting to shake" action. But for that I need just the usual verb and not a reflexive one:
Машины начали трястись.

I wrote в реке because I thought that the car and outher stuff fell in the river and the action is completed. In your version в реку it is like they are going on the way to the river if I am not mistaken.
You're not mistaken. But you were mistaken about the first thing.
Что-то упало в реку - this implies that something wasn't in the river in the first place but somewhere else but it sure fell in the river after all
Что-то упало в реке - this implies that something was in the river in the first place and it fell inside the river, perhaps deeper or something