I am currently reading up on partial derivatives and differentials in general. And there are a few points that seem unlcear to me (notation-wise). 

1. For example, if $f:\mathbb R\to\mathbb R,x\mapsto f(x)$ is a function, then is the following notation correct?
$$\frac{d}{dx}f(x)=\frac{df}{dx}(x)=\frac{\partial}{\partial x}f(x)=\frac{\partial f}{\partial x}(x)=f'(x)$$
2. Now, in one of my lectures we wrote
$$g(x):=\frac{\partial \log (f(x))}{\partial x}=\log'f(x)\cdot f'(x)=f'(x)/f(x)$$
The part about differentiating the function is clear, using the normal chain rule $(h\circ f)=(h'\circ f)\cdot f'$. What confuses me a bit is the notation since $x$ seems to have more than one meaning. Would it be "more" correct to write
$$g(x):=\frac{\partial \log (f(y))}{\partial y}\Bigg|\_{y=x}$$
as in we differentiate w.r.t. $y$ and then plug in $x$ for $y$? Also, in this particular case, could we replace $\partial$ by $d$, or would this lead to different implications?
3. And lastly, what does $df$ or $dy$ even mean? I read that the differential of $y=f(x)$ is defined to be $dy=f'(x)dx$. But how can we interpret this formula? This becomes specifically confusing when I look at the differential chain rule
$$\frac{dh}{df}=\frac{dh}{dg}\cdot\frac{dg}{df}$$
What I find strange is that one seems to be using $g$ as a function as well as a variable. How does this work exactly?