How to explain negative correlation between quantity sold and expenditure on advertisement?

In the output above, only psn is significant (based on t stats). Our Professor told us that we should consider only significant coefficients in the demand function. I am finding it difficult to agree with. In this case, if I consider only psn in my demand function, I am perhaps violating the null hypothesis, which is based on F stats (that all coefficients are 0). Also, if I consider only psn I am essentially negating the combined effect of all three variables, and basically choosing a different model that what I fitted.

It is important to take into account significance because you cannot directly observe data generating process. Regression does not tell you what $beta_0, beta_1 ... $ are, it tells you what they appear to be based on fitting line that minimizes the sum of squared residuals through data. Yes in expectations (provided all OLS assumptions hold) $E[hat{beta}] = beta$ but this result only holds in the limit as the number of observations go to infinity (see discussion in Verbeek. A Guide to Modern Econometrics or really any econometrics textbooks for more details).

In this particular case the $F$-stat (with 3 an 11 degrees of freedom) is 4.99 which is sufficient to reject the null of joint insignificance i.e. $H_0: beta_0 = beta_1 = ... beta_k = 0$ so that is not a problem.

However, you cannot reject the null hypothesis that $beta_3$ (effect of advertising) is zero. Consequently, just taking the results of the regression you present at face value (which is not necessary always appropriate in real research but might be in classroom setting), the correct answer would be "advertising has no significant effect on quantity sold, we cannot reject the hypothesis that the true effect of advertising is 0".

adv has a negative coefficient, which is against my theoretical understanding. As I have read that it has a positive correlation with the quantity demanded. How can I explain this relationship?

This could be because you have some reverse causality. Demand equations are endogenous systems and should be modeled with simultaneous equations, not just simple regression (see this explanation for how to do that in R). Using simple multivariate regression in such cases is likely going to result in biased coefficients.

Possibly when demand is low firms ramp up spending on advertising, while when demand is high and firms quickly sell their inventory they might not want to spend extra money on advertising. This creates reverse causality issue which biases the $beta_3$ estimates that no longer give you accurate effect of adv on $y$ (quantity sold). It is possible that advertising spending increases sales, but higher sales are simply associated with less spending and the second effect from sales $rightarrow$ adv simply dominates.

This being said you mention that this is an exercise imposed on you by your teacher not real research. It is possible your teacher just wants to test if you can interpret coefficient estimates properly not if you understand concepts such as reverse causality (if you did not covered this in your class then probably your teacher is not interested in discussion of that). You should ask your teacher about their expectations regarding the answer.

there's a huge difference between the levels of prices and the adv. try to use a log or ln transformation.

And I think that you could delete any variable that is not significant, or your model wouldn't be significant.