The first one is yes, the second one no. This is because a fractional linear transformation (aka a Möbius transformation) is uniquely determined by where it sends any three distinct points. Thus, we can choose to send 0, 1, 2 to 0, 1, $\infty$, respectively, and this specifies a unique transformation; but because there is a unique transformation sending 0, 1, 2, to 0, 1, 2, (namely the identity transformation), we can't also have 3 sent to $\infty$.
The Wikipedia page also describes the method for coming up with the coefficients $a$, $b$, $c$, and $d$ in the unique transformation $\frac{az+b}{cz+d}$ that sends $z_1,z_2,z_3$ to $w_1,w_2,w_3$.