Hotelling rule and the shadow prices in a peg
Partial Answer
I can give you an answer for parts (1) and (2). As for (3), I can't give an answer unless you show some work as per our site policy on homework questions.

(1)These variables are interpreted as the change in $q$ or $p$ over time. Mathematically this is the derivative, giving us the results:
$$\dot q(t)=\frac{dq(t)}{dt}$$ and $$\dot p(t)=\frac{dp(t)}{dt}$$
(2) Hotelling's rule is not given in logarithms, as explained in as the notation is explained for (1). if it was it would be indicated by the notation of $\hat p(t)$. Written concisely as:
$$\hat p(t)=\ln(p(t))$$