Why does $\sqrt{x^2}$ seem to equal $x$ and not $|x|$ when you multiply the exponents?

The rule $(x^a)^b = x^{ab}$ is only true for positive values of $x$. With negative values, you need to be much more careful.

For example, $\sqrt x \sqrt x = \sqrt{x\cdot x}$ is only true for positive values of $x$, because for negative values, the left side is not defined.

A common source of confusion or "paradoxes" comes from not paying close attention to the (perhaps rarely exercised) restrictions or boundary conditions. These restrictions are necessary to ensure that paradoxes like you're considering do not arise (i.e., otherwise the definitions would fail to be well-defined). For example, here's a proper definition of rational exponents from Michael Sullivan's College Algebra:

Note that we only consider real numbers here. Now, to answer your questions:

But since $\sqrt{x} = x^{\frac{1}{2}}$ shouldn't $\sqrt{x^2} = (x^2)^{\frac{1}{2}} = x^{\frac{2}{2}} = x$ because of the exponents multiplying together?

The first assertion is not generally true; $\sqrt{x} = x^{\frac{1}{2}}$ only provided that $\sqrt{x}$ exists (that is, not for negative $x$). In your chained equality, the second equality is false, because the exponent in $x^{\frac{2}{2}}$ contains common factors (i.e., is not in lowest terms). These statements would, however, be true if $x$ was restricted to positive real numbers only.

Also, doesn't $(\sqrt{x})^2$ preserve the sign of $x$? But shouldn't $(\sqrt{x})^2 = (\sqrt{x})(\sqrt{x}) = \sqrt{x^2}$?

All of these equalities are false for negative $x$, because in that case the expression $\sqrt{x}$ does not exist in real numbers (i.e., it's undefined). Likewise, if you look carefully at the rule for multiplying radicals, then you'll see the same restriction against square roots of negative numbers.

Edit: Added text from Precalculus: a right triangle approach by Ratti & McWaters. Hopefully this clarifies the rule for products of square roots (namely that only positive radicands can be generally combined or separated). Also, note the warning from the section on complex numbers that doing so in that case is illegitimate.