Probability and Statistics (Quartiles) Q: Find the upper and lower quartiles of the random variabl

Can anyone help me with statistics?
Question: Following is the distribution of marks obtained by 500 candidates in the statistics paper of a civil services examination:
+ marks more than: 0 10 20 30 40 50
+ number of candidates: 500 460 400 200 100 30
Calculate the lower quartile marks. If $70\mathrm{\%}$ of the candidates pass in the paper, find the minimum marks obtained by a passing candidate.
In the above problem, I am able to find the first part to calculate lower quartile marks but not able to find the second section.
Anyone, please help I am a newbie to statistics.
Thanks in advance, please ignore the bad English.

What is the meaning of percentile?
I am confused by the term percentile. Once my teacher told me that percentile means the percentage with respect to the score of the highest achiever.
This means that if in a competition I got $80$ out of $100$ and the highest score in that competition was 90 out of 100 then my percentile would be $\frac{80}{90}\ast 100=88.89$
So I got $80\mathrm{\%}$ and $88.89$ percentile.
I was believing that my above concept was right.
But when I see the definition of percentile on Wikipedia then I got something new (but I don't understand this definition) and then I thought that what my teacher told me was wrong.
Kindly tell me if my teacher right or wrong.

What is the lower quartile of the set of data?
I came across this problem asking the lower quartile of the ungrouped data. My answer is 3, but other references say it should be 2.5. Here's the data:
1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 10, 11, 14, 15, 20, 22
What do you think?

$\mu =m=\frac{q_1+q_3}{2}$
Let $X$ be a random variable with p.m.f./p.d.f. $f_X(x)$ that is symmetric about $\mu (\in R),$ i.e., $f_X(x+\mu )=f_X(\mu -x)$, $\mathrm{\forall }x\in (-\mathrm{\infty },\mathrm{\infty }).$
If $q_1,m$ and $q_3$ are respectively the lower quartile, the median and the upper quartile of the distribution of $X$ then show that $\mu =m=\frac{q_1+q_3}{2}$
How to prove.....
any Hints.

calculating upper and lower quartiles
The number of goals scored by a football team during a months worth of matches is recorderd.
They scored $0$ goals $4$ times, $1$ goal $6$ six times and $2$ goals $3$ times. Calculate the lower and upper quartiles.
Would it simply be this?: lower quartile is the $3.5$th value which is zero, and upper quartile is the $10.5$th value which is $2$?

Calculating the position of the median
I know this is a simple question, but I cannot find a straight answer anywhere.
When calculating the medium of listed data, the formula is (n+1)/2. My Statistics teacher said for grouped data the position of the medium is n/2. However, this seems contradictory as for discrete grouped data, the data could be written in a list if the original values were known. Therefore two different values for the median are found.
I know this question is probably going to get flagged as it has already been asked, however, it has never been answered. I find it frustrating that such a fundamental concept in Statistics, what is supposed to be precise and never subjective has a wishy-washy answer.
Rant over, I think the position should be (n+1)/2 for discrete data and n/2 for continuous data. Moreover, this then leads into the obvious question of how we should calculate quartiles as well.
It would be nice if everyone could agree on a certain method for calculating the medium as all my textbooks are saying different things.
Someone please dispell the confusion and this statistical mess.