Why do most character tables include an $(x^2 + y^2)$ term?
The functions on the right hand side of the table are the 'basis functions' for the irreducible representation, which are the rows of character, say $A_g$ or $B_g$ in C$_{2h}$. The basis functions have the same symmetry properties as the atomic orbitals with the same name (but other functions in $x, x^2, x^3$ etc and similarly for  $y,z$, could be used).
To determine where  an $xy$ function goes, look at a $d_{xy}$ orbital shape and subject it to each of the symmetry operations in turn,$E, C_{2}, i$ etc depending on point group.  (Assume z is the principal axis). Count +1 if the orbital (or function) is unchanged and -1 if not. So in C$_{2h}$, a $C_2$ operation leaves $xy$ unchanged as does $\sigma_h$ and $i$ so $xy$ is a 'basis' for A$_g$. The $xz$ transforms as B$_g$; a $C_2$ operation gives -1, as does $\sigma_h$, but $i$ produces +1.
(note in cases such as $z^2 -(x^2+y^2)/r^2$ the $x^2+y^2$ part is totally symmetric in the $xy$ plane and does not change anything so can be ignored; think of the donut around d$_{z^2}$.)