Lagrangian mechanics and time derivative on general coordinates

Your confusion really just comes down to understanding the notation that is widely used for partial derivatives.

For simplicity, I'll restrict the discussion to a system with one coordinate degree of freedom $x$. In this case, the Lagrangian is a real valued function of two real variables which we suggestively label by the symbols $x$ and $\dot x$. Mathematically, we would write $L:U\to\mathbb R$ where $U\subset \mathbb R^2$. Let's consider the simple example $$ L(x, \dot x) = ax^2+b\dot x^2 $$ When we write the expression $$ \frac{\partial L}{\partial \dot x}(x, \dot x) $$ this is an instruction to differentiate the function $L$ with respect to its second argument (because we labeled the second argument $\dot x$) and then to evaluate the resulting function on the pair $(x, \dot x)$. But we just as well could have written $$ \partial_2L(x, \dot x) $$ To represent the same expression. Both of these expressions simply mean that we imagine holding the first argument of the function constant, and we take the derivative of the resulting function with respect to what remains. In the case above, this therefore means that $$ \frac{\partial L}{\partial\dot x}(x, \dot x) = 2b\dot x $$ because $x$ labels the first argument, and taking a partial derivative with respect to the second argument means that we treat $x$ like a constant whose derivative is therefore $0$. It it in this sense that the partial of $x^2$ with respect to $\dot x$ is zero.

So to recap, when we are taking these derivatives, we just keep in mind that the symbols $x$ and $\dot x$ are just labels for the different arguments of the Lagrangian.

You might ask, however, "if $x$ and $\dot x$ are just labels, then what relation do they have to position and velocity?" The answer is that after we have treated them as labels for the arguments of $L$ in order to take the appropriate derivatives, we then evaluate the resulting expressions on a $(x(t), \dot x(t))$, the position and velocity of a curve at time $t$, to obtain equations of motion.

For example, if you take the example of $L$ that I started with, we get $$ \frac{\partial L}{\partial x}(x, \dot x) = 2 ax, \qquad \frac{\partial L}{\partial \dot x}(x, \dot x) = 2b\dot x $$ now we evaluate these expressions on $(x(t), \dot x(t))$ to obtain $$ \frac{\partial L}{\partial x}(x(t), \dot x(t)) = 2 ax(t), \qquad \frac{\partial L}{\partial \dot x}(x(t), \dot x(t)) = 2b\dot x(t) $$ so that the Euler-Lagrange equations become $$ 0=\frac{d}{dt}\left[\frac{\partial L}{\partial \dot x}(x(t), \dot x(t))\right] - \frac{\partial L}{\partial x}(x(t), \dot x(t)) =\frac{d}{dt}(2b\dot x(t)) - 2ax(t) $$ which gives $$ b\ddot x(t) = a x(t) $$ Once you understand all of this, you can (and should) dispense with the long-winded notation I used here for illustrative purposes, and you should make no error in using the abbreviated notation in your original post.

This is a somewhat unintuitive step in the Lagrangian formalism and the Euler-Lagrange equations. Note that you are taking the partial derivative of the Lagrangian with respect to some coordinate. Strictly speaking, when you take a partial derivative you should specify what you are holding constant.

Although we usually think of a coordinate and its time derivative as being related, when applying the Euler-Lagrange formalism we vary the generalized coordinates and velocities independently. This means that $$\frac{\partial \dot q}{\partial q} = 0, \ \ \frac{\partial q}{\partial \dot q} =0,$$ for any generalized coordinate $q$.

So, in your example, $\big(\frac{\partial L}{\partial x}\big)_{\dot x} = 0$, in fact. Here, I am using the parentheses to note explicitly that $\dot x$ is held constant.