Please correct this short story if needed. Thanks in advance.

Well written Kuba! Keep up the good work =)

Please correct this short story if needed. Thanks in advance.

She was not able to stick her neck out, so she couldn't get the fear off her chest??. > She was not willing to stick her neck out by telling him that, for she feared how her boyfriend would react. = She was not willing to take the risk of telling her boyfriend that, for ...  > She was afraid of how her boyfriend would react (upon hearing the truth) .

> I will not stick my head out for you, for you are not my friend

Furthermore/ Anyway, she had a lot of work to do that week. 

Untill next day she had to hold/ hand in her essay.>Furthermore, she had an essay to submit the next day.

Emma had planned to finish it ahead of its due date/ deadline (date),   /ahead of its  scheduled time -- which was challenging by the way, but she hadn't could not coped with it , for she found it too much of a challenge for her.

> a schedule is a timetable for a list of things that should be done at a particular point in time . The plane has arrived as scheduled. ( as planned) She is meeting a deadline given to her to finish it, not according to her schedule for doing it.

What's more, the girl/ she  had to study for her final biology exam.

She was in over her head. > She felt overwhelmed.

> The questions given to her were way over her head ( beyond her ability to understand )

In the end she dealt with everything succesfully. > Incredibly, she dealt with all of them successfully in the end.

Then she decided to go away for a few days. > After having done all that , .......... "Then" implies a time period that it far too short to connect with the nature of your previous sentence, as if she thought about it the moment she finished her exam.

Without any planning, Emma just wanted to play it by ear. > Without any planning on where to go, she just went to where her whims took her.

> Without planning how long or where to stay, Emma decided to play it by ear when she gets there.

Now she is painting the town red in LA, and she is not about to get home anytime soon.