Does $x=x$ represent a valid algebraic equation?

1. Any kind of $f(x)=g(x)$ formed things are an equation.
2. It can be simplified into $1=1$ by subtracting $x-1$ to each side.
3. Probably, it might cover the whole graph.
4. Any kind of graph, either $x$ plot or $x,y$ plot or $x,y,z$ plot must be filled and will have no space left.

1. Yes, it is a valid algebraic equation.
2. Yes, when $x\ne0,x=x$ can be simplified to $1=1$ by division. $x=x$ is always True but True is not an algebraic equation so it can't be "simplified" to True.
3. Since all real numbers satisfy $x=x$, the graph would be the entire number line in $\Bbb R$ and the entire plane in $\Bbb R^2$ if you choose to introduce another variable $y~(x+0y=x)$. Note that $y$ takes any value for any $x$ and is not dependent on $x$ as expected.

Contrast this to the equation $x+y=1$ where $y$ only takes specific values for given $x$.