Question: The direction of the gradient is along the direction of the maximum ascent, or along the directional derivative that has by the most positive value, but when talking about level surfaces, it is agued that gradient is always perpendicular to the tangent plane? This is confusing me a lot? Have I not clearly understood the difference between a level surface and a plot of a two dimensional function? Is there any difference at all?

Answer: 
Hint:
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For $m = n$ , the condition is always True.

Consider $ m \ne n $ such that : $$\frac mn = x \text{ and } \frac nm = y \space \space $$

Then $$\frac1x + x = n \space \space \text{ such that } n\in \mathbb Z$$

Can you find the values of $x$ for which $\frac1x + x $ is an integer?
Clearly, the expression does not change when we multiply $n,m$ by the same factor. Hence consider first the case that $\gcd(n,m)=1$. Then
$$\frac nm+\frac mn =\frac{n^2+m^2}{nm}=2+\frac{(n+m)^2}{nm}.$$
As $\gcd(n+m,n)=\gcd(n,m)=1$ and $\gcd(n+m,m)=\gcd(n,m)=1$, the fraction at the end is already in shortest terms, perhaps up to a unit, i.e., we need $|nm|=1$, so $|n|=|m|=1$.

Allowing $\gcd>1$ again, this amounts to $n=\pm n$ and on the other had, the expression is indeed the integer $\pm2$ if $m=\pm n$.