Q1. It means exactly what it says. :-) How much does one variable change, with respect to (that is, in comparison to) another variable? For instance, if $y = 3x$, then the derivative of $y$, with respect to $x$, is $3$, because for every unit change in $x$, you get a three-unit change in $y$.
Of course, that's not at all complicated, because the function is linear. With a quadratic equation, such as $y = x^2+1$, the derivative changes, because the function is curved, and its slope changes. Its derivative is, in fact, $2x$. That means that at $x = 1$, an infinitesimally small unit change in $x$ gives a $2x = 2$ unit change in $y$. This ratio is only exact right at $x = 1$; for example, at $x = 2$, the ratio is $2x = 4$.
This expression is the limit of the ratio $\frac{\Delta y}{\Delta x}$, the change in $y$ over the change in $x$, over a small but positive interval. The limit as that interval shrinks to zero is $\frac{dy}{dx}$.
Q2. You will rarely see, at this stage, $\frac{d}{dx}$ by itself. It will be a unary prefix operator, operating on an expression such as $x^2+1$. For instance, we might write
$$
\frac{d}{dx} \left(x^2+1\right) = 2x
$$
It just means the derivative of the expression that follows.
Q3. This is an unusual formulation. Ostensibly, though, it would mean the derivative of the operand with respect to $f(x)$, which you can obtain using the chain rule:
$$
\frac{dx}{df(x)} = \frac{\frac{dx}{dx}}{\frac{df(x)}{dx}} = \frac{1}{f'(x)}
$$
and
$$
\frac{d}{df(x)} g(x) = \frac{\frac{dg(x)}{dx}}{\frac{df(x)}{dx}}
= \frac{g'(x)}{f'(x)}
$$
Best Answer
$f'$ is a function, so $f'(2x + 1)$ denotes $f'$ applied to $2x + 1$, or $\frac{df}{dx}(2x + 1)$. For example if $f = x^2$ then $f' = 2x$ and $f'(2x + 1) = 4x + 2$.
$(f(x^2 + 1))'$ is the derivative of the function $f(x^2 + 1)$, which is $2x f'(x^2 + 1)$ by the chain rule.
This question highlights a weakness of the $'$ notation, which is that it always comes with an implied variable with respect to which you're differentiating. If this variable is clear from context there's no problem, but sometimes it isn't.