Confusion in beginner's analysis on manifolds / differential geomtry course

Question

I'm following an analysis on manifolds course and there are some concepts that I find not clear at all. Let $f, g:M \to M$ be smooth, $p \in M$ and $X$ be a vector field on $M$:

1. (Solved in comments) Given a curve $\gamma$ on $M$ such that $ \gamma(0)=p, \gamma'(0)=\textbf{v}$ define the directional derivative of $f$ along $\textbf{v}$ at $p$ as $D_\textbf{v}f (p) = \lim_{h\to0} \frac{1}{h}(f(\gamma(h))-f(p))$, which is itself apoint on $M$. What does that subtraction mean since $f$ outputs points on $M$? Is the directional derivative only defined for functions that output ub the tangent space? Is there a push-forward missing in the definition?
2. (Solved in comments) My professor wrote $X(fg)(p)=g\cdot X(f) + f \cdot X(g)$. What does that multiplication mean since $f$ and $g$ output points on $M$?
3. We said that $X(f)(p) := D_{X(p)} f(p)$. Is it just me or this notation is terrible? Because $X$ is also defined as a thing that eats a point $p$ on $M$ and spits a vector in $T_pM$, so one would naturally indicate $X \circ f $ as $X(f)$, but when we apply this to a point we mean a completely different thing, namely $X(f(p))$ (a vector in $T_{f(p)}M$) vs $D_{X(p)}f(p)$ (a point in $M$). Am I missing something here?
4. When we wanted to represent the Lie derivative in coordinates we started by (and I quote) observing that $$f_* \frac{\partial}{\partial x^j}(x) = \sum_l \frac{\partial f}{\partial x^j}\frac{\partial}{\partial x^l}(f(x))$$ where $f_*$ is the push-forward of $f$ and $\frac{\partial}{\partial x^j}$ are the basis vectors given by some parametrization $\phi$. I've got no idea what this means, why would $\frac{\partial}{\partial x^j}$ take in any arguments if it is simply defined as $T\phi(e_j)$? $f_*$ is a function $:TM \to TM$, so $f_*\frac{\partial}{\partial x^j}$ should already be defined, what does that $x$ mean? Could it be that he wanted to indicate that $\frac{\partial}{\partial x^j}$ was a basis vector at the point $x$? I thought the standard notation for this was $\left.\frac{\partial}{\partial x^j}\right|_{x}$
5. This is just a confirmation, we define the Lie derivative of a vector field $Y$ along $X$ as $$ \mathcal{L}_XY (p)= \lim_{h \to 0} \frac{Y(p) - (TF_X^h)(Y(F_X^{-h}(p)))}{h}$$ where $F_X^h:M \to M$ is the flow of $X$. If I understand this correctly then the term $(TF_X^h)(Y(F_X^{-h}(p)))$ is to be interpreted as moving slightly away from $p$ (the inner $F_X^{-h}(p)$ bit), applying $Y$ to it, obtaining a vector in $T_{F_X^{-h}(p)}M$ and then translating the resulting vector back in $T_pM$ via $TF_X^{h}$ so that we can confront this with $Y(p)$. Is this the purpose of that $TF_X^h$ bit? So that $Y(p)$ and the other vector live in the same space and we can perform the subtraction?
6. When defining the Lie Bracket we come across the composition of vector fields; what does that even mean? I can understand this when we define it on functions as $X(Y(f))(p) = D_{X(p)}D_{Y(p)}f(p) = \mathcal{L}_X(D_{Y(p)}f)(p)$ but on points it just doesn't make any sense since $Y(p)$ is a tangent vector and $X$ takes points in input.

Sorry for the long post but I feel like I have a lot of confusion in my head and I can't find many resources that follow the same path as my profressor; normally I'd ask my colleagues but due to covid my university is currently closed. I'll probably add more questions in the future.

Bonus: I get that these concepts are the natural generalizations of those found in $\mathbb{R}^d$, but I can't really see them in an intuitive way. Like, how can I visualize a map from a manifold to another one? And its derivative? We proved that $\mathcal{L}_XY = 0 \iff (F_X^h)_*Y = Y$ which I feel should give me a grasp of what this means, but it's currently beyond me.