In the equation
$$\frac{n-1}{n(n-1)(n-2)!} = \frac{1}{n(n-2)!} $$
can I cancel out the factors $(n-1)$'s in the numberator and denominator, so the equation is equal?
I've learned that you can't cross anything if there is a + or - sign in the fraction?
Yes the cancellation is allowed
$$\frac{n-1}{n(n-1)(n-2)!} = \frac{1}{n(n-2)!}$$
provided that we exclude case $n=1$ from the solutions.
For example the the following equation
$$\frac{(x-1)^2}{(x-1)}=0 \stackrel{x-1\neq0}\implies x-1=0$$
has not solution because we need to exlude the case $x=1$.