- what does the passage means
I hope it will be clear from my answers to the other 6 questions.
2-3. how tensor changes under proper or improper rotation
By rotations we mean Lorentz transformations. Spatial rotations are just special cases of Lorentz transformations. An arbitrary Lorentz transformation is given by a $4 \times 4$ matrix $\Lambda_{\mu}^{\;\nu}$, which satisfies the following condition:
$$ \eta_{\mu \nu} = \eta_{\mu' \nu'} \; \Lambda_{\mu}^{\;\mu'} \Lambda_{\nu}^{\;\nu'}. $$
This is equivalent to saying that the matrix, when applied to a 4-vector, does not change its invariant interval.
Note that if we take $\det$ from both sides, we get
$$ (\det \Lambda)^2 = 1 \quad \Longrightarrow \quad \det \Lambda = \pm 1. $$
Lorentz transformations with positive (negative) determinants are called proper (improper) respectively.
Tensors transform under Lorentz transformations according to
$$ A_{\mu \nu \dots \sigma} \rightarrow A_{\mu' \nu' \dots \sigma'} \; \Lambda_{\mu}^{\;\mu'} \Lambda_{\nu}^{\;\nu'} \dots \Lambda_{\sigma}^{\;\sigma'}. $$
4-5. how pseudotensor changes under proper or improper rotation
Pseudotensors transform under Lorentz transformations according to
$$ B_{\mu \nu \dots \sigma} \rightarrow \det \Lambda \cdot B_{\mu' \nu' \dots \sigma'} \; \Lambda_{\mu}^{\;\mu'} \Lambda_{\nu}^{\;\nu'} \dots \Lambda_{\sigma}^{\;\sigma'}. $$
- how to decompose an improper rotation into the product of rotation and reflection
Consider, for example, an (improper) spatial reflection, given by
$$ \Lambda_{\text{refl}} = \left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{array}\right). $$
Any improper Lorentz transformation can be written as
$$ \Lambda_{\text{improper}} = \Lambda_{\text{proper}} \cdot \Lambda_{\text{refl}}, $$
where $\cdot$ is matrix multiplication.
- det(rotation x reflection) = -1. Does det R' = -1 implies R' is an improper rotation?
Exactly, it is the definition.
