It is not particularly important that the right hand side is one, but it is important that the right hand side is bounded by two different numbers (such as $1$ and $-1$) in order to induce a margin between the clusters. As you note, "we can adjust for this number". For instance, $w^Tx_i + b = 2$ is equivalent to $[\frac 12 w]^T x_i + \frac 12 b = 1$ and to $w^Tx_i + (b - 1) = 1$.

It is difficult to solve the norm $||w||$ because that involves square root. That's why we can square this result to $||w||^2$ without any problem.

And we usually append $\frac{1}{2}||w||^2$ for mathematical purpose when we will derivate the function for optimize it using Lagrange multiplier to find the solution.