The reason for why you can cancel out the term $x - 4$ is that you aren't calculating the value of the function at point $x = 4$. If you were calculating the value of the function at point $x = 4$ you would be dividing by zero, an undefined operation.
If you see the function as a machine where $x$ is inputted in one side while the output is $f(x)$, then comes the question: "What if $x = 4$? Division by $0$?" The answer is that $4$ is not part of the function's domain, which means that $x = 4$ isn't accepted as input and therefore nothing happens. The limit tells you that for $x$ extremely close to $4$, $f(x)$ is extremely close to $8$.
To given an analogy: imagine the function as a box with two holes, one to put things in, another were things come out. The number $4$ is not part of the function's domain, which in this case is something with some shape that won't go in the box, it won't pass through the hole.
You factorize or multiply by a conjugate because you are replacing an expression with something equivalent. In calculus you take advantage of properties from real numbers and algebra, many properties, but you don't prove them.
The limit of a function doesn't exist when the right and left sides approach different quantities. The usual cases are picewise functions and rational functions where there is a vertical asymptote.