I Scaling Dimension of a Field in CFT

[shinobi20]

TL;DR

I'm quite unsure about how the scaling dimension of a field is devised. I need clarifications on small details in order to make sure that my concepts are clear.

I'm studying CFT, and I find the lecture notes and books really confusing and devoid of explanations (more details).

In a scale transformation $x' = \lambda x$, the field $\phi(x)$ should also be affected by the scale transformation, i.e., $\phi'(x') = \phi'(\lambda x) = \lambda^{-\Delta} \phi(x)$. I think this should mean that we assume that the field should scale but we do not know by how much, and $\Delta$ quantifies this unknown, which is called the scaling dimension. We put a minus sign because when we plug it in the action, this will avoid a negative dimension (see below)?

If the scale transformation is a symmetry of the theory, then the action must be invariant under this. Particularly, in a free theory,

\begin{align*}
S' & = \int d^d x' \partial'_\mu \phi'(x') \partial'^\mu \phi'(x')\\
& = \int d^d x' g^{\mu \nu} \partial'_\mu \phi'(x') \partial'_\nu \phi'(x')\\
& = \int d^d x \Bigg| \frac{\partial x'}{\partial x} \Bigg| \lambda^{-2} g^{\mu \nu} \partial_\mu \phi'(x') \partial_\nu \phi'(x')\\
& = \int d^d x \lambda^d \lambda^{-2} \partial_\mu \phi'(x') \partial^\mu \phi'(x')\\
\end{align*}

Comparing this with $S = \int d^d x \partial_\mu \phi(x) \partial^\mu \phi(x)$,

\begin{equation*}
\int d^d x \lambda^d \lambda^{-2} \partial_\mu \phi'(x') \partial^\mu \phi'(x') = \int d^d x \partial_\mu \phi(x) \partial^\mu \phi(x)
\end{equation*}

We can infer that,

\begin{equation*}
\lambda^{\frac{d-2}{2}} \phi'(x') = \phi(x), \quad \text{OR} \quad \phi'(x') = \lambda^{\frac{-(d-2)}{2}} \phi(x)
\end{equation*}

So the scaling dimension is $\Delta = \frac{(d-2)}{2}$. If we were to not put a minus sign from the start, i.e., $\phi'(\lambda x) = \lambda^{\Delta} \phi(x)$, then $\Delta = \frac{-(d-2)}{2}$.

Questions:
1. Can anyone verify if what I'm saying (my statements) above are correct?
2. Can anyone explain more about the minus sign?
3. I'm not sure if what I wrote in the second line of the action is correct, i.e., $\partial'_\mu \phi'(x') \partial'^\mu \phi'(x') = g^{\mu \nu} \partial'_\mu \phi'(x') \partial'_\nu \phi'(x')$. Is this correct or should it be $\partial'_\mu \phi'(x') \partial'^\mu \phi'(x') = g'^{\mu \nu} \partial'_\mu \phi'(x') \partial'_\nu \phi'(x')$? If this is the case then $g'^{\mu \nu}$ should also transform, but then I would not get the correct scaling dimension.