Understanding the metric transformation under infinitesimal diffeomorphism

In my general relativity course, we are discussing infinitesimal diffeomorphisms defined by $x^{mu}rightarrow y^{mu}(x) = x^{mu} + xi^{mu}(x)$. We have been examining how different objects transform under this. For example, the metric transformation $g_{munu}(x)dx^{mu}dx^{nu} rightarrow g_{munu}(y) dy^mu dy^nu$ can be found via

$$g_{munu}(y) dy^mu dy^nu = g_{munu}(x+xi)frac{partial(x+xi)^mu}{partial x^alpha} frac{partial (x +xi)^nu}{partial x^beta} dx^alpha dx^beta tag{1} $$

From here, it is my understanding that we Taylor expand the expressions to get:

$$ = left( g_{munu}(x) + xi^alpha partial_alpha g_{munu}(x),+, … right)(delta^mu_alpha + partial_alpha xi^mu ) (delta^nu_beta + partial_beta xi^nu,) mathrm d{x^alpha} mathrm dx^beta tag{2}$$
$$= left( g_{munu}(x) +, xi^sigma partial_sigma g_{munu}(x) +, g_{musigma} partial_nu xi^sigma +, g_{nusigma} partial_mu xi^sigma,+,… right) mathrm dx^mu mathrm dx^nu tag{ 3}$$

where the dots are just higher order terms that we can ignore. From here, one can show that the metric varies with the Lie derivative in the direction of $xi^{mu}$.

I am really struggling to comprehend the mathematical calculation and simplification in this problem. My specific questions are:

Q1) From (1) to (2), how does one taylor expand an expression like $partial(x+xi)^mu / partial x^alpha$? I am having trouble understanding how one gets to the resulting expression.

Q2) From (2) to (3), I am completely lost on what was done. It seems like the kronecker delta’s were applied to $mathrm dx^alpha mathrm dx^beta$. I assume they would need to be factored out of the binomial expressions, but how does that affect the other terms?

Q3) I am often confused when new indices are relabeled. What is the motivation for introducing this sigma index?

Q4) On a more conceptual note, I understand that we want to use this result to show that the action is invariant under infinitesimal diffeomorphisms. But why look at infinitesimal diffeomorphisms rather than diffeomorphisms in general?

I apologize if much of this question is trivially mathematical. I debated putting it in the Math forum but figured I’d rather get a physicist’s answer.