Let us assume that we have constructed a $G$-principal bundle $P$ over the manifold $M$ (for a curved space-time this is a $GL$-bundle, for a gauge theory I take $U(1)$ = electrodynamics) and the corresponding associated bundle $P_F$ with a typical fibre $F$ being a vector space.
1) At first, I am confused about the meaning of a local section $\sigma: U \rightarrow P$ on the principal bundle, where $U \subset M$. I understand it as assigning some "point of reference" with respect to $G$ to the corresponding point in $M$. This can be seen by the induced local trivialization which sets $\sigma(x) = (x, e)$ so that the section always corresponds to a neutral element of $G$. The associated bundle $P_F$ is constructed as the set of equivalence classes of $P \times F$ with respect to the equivalence relation $(p, f) \sim (pg, g^{-1}f)$, which means that the simultaneous transformation of the basis and the components does not change the vector. Then the section $\sigma$ fixes a representative in each equivalence class in $P \times F$, and this is interpreted as fixing the frame/gauge, is this correct?
2) If so, how does a section on the associated bundle $\psi: U \rightarrow P_F$, which is some matter field for the gauge $U(1)$-bundle $P$, look like? If I assume that $\sigma$ picks up different elements of $U(1)$, does it mean that $\psi (x)$ has different phases when I go through $U$ so that I have something like $\psi(x) e^{i \theta(x)}$? For me this sounds like a mistake because this is already a gauge transformation since the phase $\theta$ depends on the point on the manifold.
3) The connection form $\omega$ acts on the tangent to $G$ components of a tangent vector $X_p$ on the principle bundle. What is the intuition behind tangent vectors in $P$?
4) Furthermore, the connection form gives a separation of tangent spaces in $P$ into the vertical and horizontal spaces, which I see intuitively as "parallel" to $G$ and $M$, correspondingly. Since I lack intuition for this choice and its relation to the local sections on $P$ I would like to consider the following example. Let us consider a flat 2-dimensional manifold with a unique chart with polar coordinates which correspond (?) to some section $\sigma$ in $P = LM$. Will it be correct to say that taking horizontal spaces $H_p P$ in such a way that tangent vectors to $\sigma(x)$ always lie in $H_p P$ means that the parallel transport of the vectors will consist in projecting the vectors to the coordinate lines during the transport, so that the corresponding connection coefficients are zero?