quantum mechanics - Queries about rotational groups $mathrm{SO}(3)$ and $mathrm{SU}(2)$ in QM

In a QM text I am using (Sakurai 2nd edition 'Modern Quantum Mechanics'), he describes two rotation groups, namely the $\mathrm{SO}(3)$ rotation group and $\mathrm{SU}(2)$ rotation group (unitary unimodular group).

He defines $\mathrm{SO}(3)$ as a group with matrix multiplication on a set of orthogonal matrices (which are matrices which satisfy $R^TR = 1 = RR^T$), he then states that this group only includes rotational operators (and not also inverse operators which would be the group $\mathrm{O}(3)$). He does not ever rigorously define 'rotational operation'.

1. How you would distinguish between rotational operators and inverse operators, would a sufficient definition be that rotational operators is a transformation with one fixed point?

He also defines the the group $\mathrm{SU}(2)$ which consists of unitary unimodular matrices, and states that the most general unitary matrix in two dimensions has four independent parameters and it is defined as

$$U = e^{i \gamma} \left( {\begin{array}{cc} a & b \\ -b^* & a^* \\ \end{array} } \right)$$ where $|a|^2 + |b|^2 = 1,~~~\gamma^* = \gamma.$

2. Am I right to assume that the $\mathrm{SO}(3)$ rotation group does not have much of application in quantum mechanics but is rather used more in classical mechanics whereas $\mathrm{SU}(2)$ is used more in quantum mechanics, particularly for $s =\frac{1}{2}$ spin systems where we work in a two dimensional Hilbert space?


3. How does it follow that there are four independent parameters for the general unitary matrix, the way I see it there are three independent parameters, namely, $a$, $b$ and $\gamma$?