Mathematical bases for which $q$ and $\dot{q}$ could be treated as independent variable in $L(q,\dot{q},t)$.
In Lagrangian mechanics with single degree of freedome $q(t)$ and it's first degree deritative $\dot{q}(t)$ were considered as independent variable.
$\displaystyle \frac{\partial \dot{q}}{\partial q}=\frac{\partial q}{\partial t\partial q}=\frac{\partial q}{\partial q\partial t}=\frac{\partial C}{\partial t}=0$ and its general form $\displaystyle \frac{\partial q^{(n)}}{\partial q}=0$ could be easily proven.
However, $\displaystyle \frac{\partial q}{\partial \dot{q}}=0$ was not so straight forward.
My question was that:
Prove the general form $\displaystyle \frac{\partial q}{\partial q^{(n)}}=0$ in Lagrangian equation $L(q,\dot{q},t)$ where $\displaystyle\frac{\partial L}{\partial q}=\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}})$ and $q^{(n)}$ was the $n$ th deritative of $q$.
As I commented on an answer to a related question, I think part of the problem is conflation of functions with variables.
If variables $x$ and $y$ are related by $y=f(x)$, that doesn't mean that $y$ is the function $f$; the same variable could be a different function of another variable, like $y=f(x)=g(z)$. A function is a relation, while a variable is... (I haven't seen a satisfactory definition.)
If we insist that the Lagrangian is a function, not a variable, then $\frac{\partial L}{\partial q}$ is an abuse of notation, and the Lagrangian equation should be
$$L^{(1,0,0)}(q(t),\dot q(t),t) = \frac{d}{dt}L^{(0,1,0)}(q(t),\dot q(t),t)$$
assuming that $t$ is a variable. Note that $L^{(1,0,0)}$ is a function in itself, regardless of what we call its inputs.
If the Lagrangian is a variable, then the partial derivatives must specify what other variables are held constant for the differentiation, as joriki said.
But how can we "hold constant" a dependent variable? (Not only is $\dot q$ dependent on the function $q$, but also the variable $q$ is dependent on $t$.) Perhaps the idea of "dependent/independent variables" doesn't make sense, and should be replaced with "domain and codomain of functions". Other views are welcome.