Would I be a full time student if I was in highschool for 5 months but I also worked a job after school

Yes you were a student.  But those questions about being a student lead to questions about college tuition, so don't let that get you confused.  You will not need to answer any of those questions.

If you are a high school student, make sure in My Info you say that you can be claimed as someone else's dependent.  You can seek a refund of the amounts in boxes 2 and 17 of your W-2, but answer that question about being a dependent correctly.

Q.  Would I be a full time student if I was in highschool for 5 months but I also worked a job after school?

A.  Yes. 

But, for most HS students that question is only academic.  Just being under 19, student or not, means you can be your parent's dependent.

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test.

A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled
2. He did not provide more than 1/2 his own support. Scholarships are excluded from the support calculation
3. He lived with the parent (including temporary absences such as away at school) for more than half the year

So, it doesn't matter how much he earned. What matters is how much he spent on support. Money he put into savings does not count as support he spent on himself.

The support value of the home, provided by the parent, is the fair market rental value of the home plus utilities & other expenses divided by the number of occupants.

With the tax law change, effective 2018, most dependents will get the same refund whether they claim themselves or not. The personal exemption has been eliminated and the standard deduction increased. However, you only qualify for an education credit or deduction, if you are not a dependent.