Does $\Gamma(r)$ in the probability density of Gamma Distribution
$ f_X(x,\lambda,r) = \frac{(\lambda \cdot x)^{r-1} \lambda e^{-\lambda x}}{\Gamma(r)}$
always equal to $(r-1)!$ ?
If so, why not just replace $\Gamma(r)$ with $(r-1)!$ in the function?
No that is true for $r>0$ which are integers. Generally $$ \Gamma(r)=\int_{0}^\infty x^{r-1}e^{-x}\, dx \quad (r>0) $$ In particular $\Gamma(1/2)=\sqrt{\pi}$.