The question asks me to determine if the following arguments are valid.
$\begin{array}{rcr} A) & \forall x \forall y K(x, y) \, \vDash \forall x K(x,x) \\ 1. & \forall x \forall y K(x,y) \\ 2. & \forall y K(x,y) & U.I\; 1. \\ 3. & K(x,x) & U.I\; 2. \\ 4. & \forall x K(x,x) & U.G\; 3. \end{array}$
Can I use universal instantiation to introduce x in that way? Is that legal? I have yet two other problems in this question and all use binary predicates like this.
There is no problem with this kind of universal instantiation.
The only thing that you need to avoid is situations like: $$\forall x \exists y: R(x,y) \leadsto \exists y: R(y, y)$$ (You can take $R(x,y)$ to be $x < y$ in $\Bbb N$ to see this is false.)
This amounts to saying that the variable you substitute for $x$ must be free (that is, not be quantified over) in all positions where $x$ occurs.