Will inverse functions, and functions always meet at the line $y=x$?

If I have a function, the inverse function, by definition will be a reflection of the original function in the line $y=x$, so if I wanted to find the point of intersection, instead of solving it with equating both of the functions to equaling each other, could I assume that the point of intersection, between the two functions, will always be at the point $y=x$?

This would enable me to solve problems much easier, instead of having to solve quartic equations, for example.

Not necessarily. If $f(x)=-x$, this implies that $f^{-1}(x)=-x$, (since $f(f(x))=f(-x)=x$), so the graphs of $f$ and $f^{-1}$ intersect everywhere.

Another example is $g(x)=-\frac{1}{x}$, which is also its own inverse, but doesn't intersect $y=x$ at all.

As already pointed out by sky90 and Marra in the comments, in general a function and its inverse do not need to have an intersection. This can be seen from the example given in the comments. Another example would be $f(x)=\exp(x)$ and its inverse $f^{-1}(x) = \log(x)$, whose graphs never intersect.

Note that apart from the case mentioned by Chappers, where a function is its own inverse and there is an infinite number of intersections, you can also find examples with a finite set of intersections, e.g. $f(x) = -x^3$ and its inverse $f^{-1}(x) = -\sqrt[3]{x}$, whose graphs intersect at the points $(-1,1)$, $(0,0)$, $(1,-1)$, where the first and the latter are clearly not on the line $y=x$.