Basic division and multiplication We know that 2 x 2 /2 can be solved by removing '2' from the denominator and numerator, we can't do the same if the operation was addition, These "rules" have been established based on the understanding of these operations and how these interact with each other.
Up until now, I've just been accepting it as a rule, but this causes a lot of discomforts while doing math.
I generally get good grades in math, but that's only by following these rules and patterns but there is no genuine understanding as to why we can remove "2" from the numerator and denominator in 2 x 2 /2 but we cannot in 2 + 2/2.
Secondly, 2 x 2 / 2 can be solved by removing "2" from the numerator and the denominator and it gives the same result as 4/2, how?.
Is it an axiom?
I hope to gain an intuition for the above concepts, so I can be free from my long-held guilt.
Thanks.

I'm not quite sure how advanced in mathematics you are so this is going to be the real simple explanation. 
The fraction bar is notation for division. $\frac{3}{4}$ represents the  number that is defined as the result of the operation $ 3\div 4.$ 
Therefore $$ \frac{2}{2} = 1$$ because $2 \div 2 = 1.$
In your case
$$ 2 \times \frac{2}{2} = 2 \times 1 = 2$$
So it looks like we just "removed" the two's. This only works because we end up by multiplying by $1$ so it is in general best not to talk about it like that. 
This obviously doesn't work for addition 
$$ 2 + \frac{2}{2} = 2 + 1 = 3.$$
Also $$ \frac{4}{2} = 4 \div 2 = 2 \quad \text{ so } \quad  2 \times \frac{2}{2} = \frac{4}{2}.$$ 
