Fokker-Planck equation

I'm struggling to proof the Fokker-Planck equation.

Let $$b:[0,T]\times \mathbb{R}^N\to\mathbb{R}^N$$ and $$\sigma:[0,T]\times \mathbb{R}^N\to\mathbb{R}^{N\times d}$$ two measurable functions.

Let $$X=\begin{Bmatrix} X_t \end{Bmatrix}_{t\in[0,T]}$$ a multivariate Ito's process with dynamic $$dX_t:=b(t,X_t)dt+\sigma(t,X_t)dW_t$$ and $$b(t,X_t)=N\times 1$$ , $$\sigma(t,X_t)=N\times d$$.

I know for hypothesis that $$\forall(t,x)\in S_t\subseteq \mathbb{R}^N,X_t^{t,x}$$ is solution for $$dX_t$$ such that $$X_t^{t,x}=x$$ in $$(\Omega,F,\begin{Bmatrix} F_t \end{Bmatrix}_{t\geq 0},\mathbb{P})$$. First problem: what's the meaning of this condition? $$t,x$$ on apex means that the solution $$X_t$$ for $$dX_t$$ depends on the time and the process itself?

Then, let $$f(X_t)\in C^2(\mathbb{R}^N)$$ a continuous and 2-derivable function (with derivatives defined on $$\mathbb{R}^N$$) with multivariate Ito's dynamic $$df(X_t)=\bigtriangledown f(X_t)\sigma(t,X_t)dW_t+\frac{1}{2}[\sum_{i,j=1}^{N}c_{i,j}(t,X_t)\frac{\partial^2f(X_t)}{\partial X_t^i \partial X_t^j}+\sum_{j=1}^{N}b_j(t,X_t)\frac{\partial f(X_t)}{\partial X_t^j}]dt$$.

Since $$f$$ depends only on the process and not on the time, we have not the spatial derivative of the time. Furthermore, $$c_{i,j}$$ is the i,j-th component of the quadratic and symmetric matrix that we obtain by multiplying $$\sigma(t,X_t)=N\times d$$ with the transposed $$\sigma^*(t,X_t)=d\times N$$.

Let also $$A_t:=\frac{1}{2}[\sum_{i,j=1}^{N}c_{i,j}(t,X_t)\frac{\partial^2}{\partial X_t^i \partial X_t^j}+\sum_{j=1}^{N}b_j(t,X_t)\frac{\partial}{\partial X_t^j}]$$ the characteristic operator of SDE $$dX_t$$. So $$df(X_t)=A_tf(X_t)dt+\bigtriangledown f(X_t)\sigma(t,X_t)dW_t$$.

So, we have the PdC $$\left\{\begin{matrix} Au-au+\frac{\partial u}{\partial t}=f\\ u(T,\cdot)=\phi \end{matrix}\right.$$ with $$f,a,\phi$$ known functions.

Good. Now, Feynman-Kac THM says that $$u(t,x)=\mathbb{E}[e^{-\int_{t}^{T}a(s,X_s)ds}\phi(X_T)-\int_{t}^{T}e^{-\int_{s}^{t}a(r,X_r)dr}f(s,X_s)ds]$$ is solution of PdC under the regularity hp. for $$b$$ and $$\sigma$$. Second problem: i don't know why.

Then, let $$F:=A_t+\frac{\partial}{\partial t}$$ with solution $$P$$. If $$\phi$$ is continuous and derivable it's shown that $$u(t,x)=\int_{\mathbb{R}^N}\phi (y)P(t,y|T,x)dy$$. Third problem: i don't know why.

By placing $$a=f=0$$ we obtain that $$u(t,x)=\mathbb{E}[\phi(X_T^{t,x})]$$ (with $$t,x$$ that I still don't understand).

So, it's shown (Fourth problem: i don't know why) that if exists a solution for $$F:=A_t+\frac{\partial}/{\partial t}$$, that solution is the transition density of $$dX_t$$.

In conclusion we obtain (Fifth problem: i don't know why) that the F-P equation is $$\frac{\partial P}{\partial t}=\frac{1}{2}\sum_{i,j=1}^{N}\frac{\partial}{\partial X_i \partial X_j}(c_{i,j}P)-\sum_{j=1}^{N}\frac{\partial}{\partial X_j}(b_jP)$$.

I apologize for all this unclear points but my text is too dispersed.

I really hope that some of you can help me. Thanks in advance!