Funky Functional Equation

You really need to restrict to positive integers here, since $f(0) = 0$ can be deduced from the equations (and there are a lot of negative integers that satisfy $f(n) =2n$ as well...)

Clearly $f$ (when restricted to nonnegative integers) is just the function that switches the $1$ s and $2$ s in the ternary expansion of an integer. It's not hard to see that $f(n) = 2n$ if and only if $n$ 's ternary expansion consists of only $0$ s and $1$ s. The restriction on $n$ means that it has at most $7$ ternary digits. So the total number of such nonnegative $n$ is $2^7$ , or $2^7-1 = \fbox{127}$ if we restrict to positive integers.