I am reading an excerpt from Morita's book. The story is that if you have a connection $\nabla$ on a vector bundle $E$ over some manifold $M$ and a function $M_n(\mathbb{R})\to \mathbb{R}$ which is a homogeneous polynomial invariant under conjugation, then you obtain a form on $M$ by applying $f$ to the curvature matrix $\Omega$. What needs to be shown is that $f(\Omega)$ is a closed form, and that $[f(\Omega)]$ doesn't depend on the connection you started with. The second part is what I'm having trouble with.
What Morita does is suppose we are given connections $\nabla^0$ and $\nabla^1$ on $E$. Consider the bundle $E\times\mathbb{R}\to M\times\mathbb{R}$ and define a connection $\tilde{\nabla}$ on $E\times \mathbb{R}$ in the following way. Suppose $s$ is a section of $E\times\mathbb{R}$ which is independent of $t$ (that is, $s(p,t) = s(p,t')$ for all $t, t'$). Then define $\tilde{\nabla}_{\frac{\partial}{\partial t}} s = 0$ and $$ \tilde{\nabla}_X s = (1-t)\nabla_X^0 s + t\nabla_X^1 s $$ for $X\in T_{(p,t)}(M\times\mathbb{R})$.
I understand that every vector field on $M\times \mathbb{R}$ can be written as a linear combination with function coefficients of $\frac{\partial}{\partial t}$ and vector fields which are tangent to $M\times \{t\}$. Morita says that also every section of $E\times \mathbb{R}$ can be written as a linear combination with function coefficients of sections which are independent of $t$. I don't see why this is true.
Moreover, aren't you supposed to be feeding connections vector fields and not tangent vectors? Is the point here that a vector field on $M\times\mathbb{R}$ which is tangent to $M\times \{t\}$ can be considered as a vector field on $M$?
Regarding your first question, it may help to note that the vector bundle $E\times\mathbb{R}\to M\times\mathbb{R}$ is the pullback bundle of $E\to M$ along the projection $p:M\times\mathbb{R}\to M$: $$\begin{array}{ccc}E\times\mathbb{R}&\to&E\\\downarrow&\quad&\downarrow\\M\times\mathbb{R}&\xrightarrow{p}&M\end{array}\quad.$$ In particular, if $e_1,\ldots,e_n$ is a local frame of $E$ on $U\subset M$, then it is a frame of $E\times\mathbb{R}$ on $U\times\mathbb{R}$, as well.
Regarding your second question, it is a fundamental fact that if the vector fields $X,X'\in\mathfrak{X}(M)$ satisfy $X(p)=X'(p)$ for some $p\in M$, then for a section $s$ of $E$ we have $$\nabla_Xs(p)=\nabla_{X'}s(p).$$ (This can be shown in a few different ways, depending on your definition of a connection). Hence, the expression $$\nabla_vs$$ is actually well-defined for a tangent vector $v$.