Is this OK? Here is a link to the first page of a proof in Mathematics Magazine. There is also this video on YouTube. If you type
trapezoid rule error proof
into Google, you get these, and more.
BaronVT already pointed out an error in your estimate for $-4\sin{2\xi}+6\sin(\xi)+2(\xi)\sin(\xi)$: the estimation process is not as simple as "replace all $\sin$ and $\cos$ with $1$". E.g., $\sin 2\xi\le 1$ does not imply $-4\sin 2\xi\le -4$; rather it's $-4\sin 2\xi\ge -4$. With the help of triangle inequality you can get a correct bound.
To your main question
How do I know if I have the correct bound?
There is no such thing as the correct upper bound. There is the smallest one (infimum), but it's usually impossible to find without knowing the exact result of integration -- and if we knew that, why would we need a numerical method in the first place?
If $0.02444$ is an upper bound for the error, then so are $0.03$, and $0.7$, and $42$, and $10^{78}$ -- these are upper bounds too. They are no less correct than $0.02444$. Some of them may be less useful, or in fact totally useless. But usefulness is somewhat subjective; there is no mathematical definition of it.
In practical terms: if you are taking a course in numerical analysis, then correct means "what your professor expects of you" (within some margin), and your indicator that an answer is correct is "it's close to the answer at the end of the book".
If an estimate calls for an upper bound on, say, $x\sin x+x^2\cos x$ on $[0,\pi/2]$, then it's reasonable to do
$$x\sin x+x^2\cos x\le x+x^2 \le (\pi/2)+(\pi/2)^2$$
But someone with more patience and energy can spend a few more minutes and come up with a better (smaller) bound. Or spend an hour and come up with a yet better bound. Or spend a day... you get the idea. What you need is a reasonably good bound obtainable with reasonable amount of effort, that's all.
Best Answer
So, we need $n\ge \left(\frac{6\cdot 10^3}{12}\right)^{\frac12}\approx 22.4$. What other restrictions are there on $n$? Just that it's an integer. The smallest integer $n$ satisfying the inequality is $23$. Then that gives $h=\frac1{23}$ for the answer to the original question.