First, let's take a look at your objective function:
$$
\begin{align}
f(x)&=(y-Gx)^{T}(y-Gx) \\
&=y^Ty-y^TGx-x^TGy+x^TG^TGx \\
&=y^Ty-y^TGx-y^TGx+x^TG^TGx \\
&=y^Ty-2y^TGx+x^TG^TGx \\
\end{align}
$$
This is a quadratic function, and we can put it in a more standard form as follows:
$$
\begin{align}
a&=y^Ty \\
b^T&=-2y^TG \\
b&=-2G^Ty \\
C&=2G^TG \\
f(x)&=a + b^Tx + \frac{1}{2}x^TCx \\
\end{align}
$$
Then you offer two possibilities for constraints: inequality constraints, or linear inequality constraints.
Calculus tells us that for a bounded optimization problem, the optimum value will be either a point in the interior of the permissible region at a spot where the gradient of the function is zero, or on the bounds of the permissible region.
This is pretty complicated to do explicitly, and not a viable strategy to code specifically for your problem unless $n\le2$.
All of this is really just to back up the following claim about your question (2): sadly, there is no nice simple formula like in the unconstrained case.
Fortunately, your problem fits into a common and well-researched class of problems: quadratic minimization with linear inequality constraints, which is a subset of quadratic programming. Thus, to answer (1) and (3), you simply need to look up and implement one of several well-known algorithms (a decent reference I googled on the topic: http://www.math.uh.edu/~rohop/fall_06/Chapter3.pdf), or use an existing implementation for your programming language/environment of choice. Examples include:
Googling "[your language name] quadratic programming" will likely give you some useful stuff for nearly any common language. Using an existing implementation is a much better idea than writing your own, it's pretty hard to get right.
Finally, if you can't find a specialized quadratic optimizer, you may find a general nonlinear optimizer that takes inequality constraints. It will be a performance hit relative to a specialized quadratic algorithm, but unless your problem is huge, or you have a very tight computation time budget, it will probably get the job done.