There need not be any patterns at all.
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The denominator will change for some operations and not for others.
some of them denominatar are divisible by numerator and some or not
If the denominator is zero at some point, then the function is not defined at the corresponding points.
No, there are no ongoing patterns. There are some micro-patterns inside pi (such as 123456789) but they have no meaning.
Only after simplification. If you subtract numbers with the same denominator, you have to put the original denominator in the result, but you can later simplify this, in some cases. A simple example: 3/4 - 1/4 = 2/4. Same denominator, but this can be simplified to 1/2.