On Question 1, you have zero delta (since you delta hedge) and negative gamma. So the cases where you lose money are (1) large upward movement of the underlying, (2) large downward movement of the underlying. The cases where you make money are small upward or downward movements. Another way to look at it is that you lose money in case of an increase in vol, make it with a decrease in vol.
On Question 2, if you are long a put you have negative rho. You can hedge this with some positive rho, from a short put or a long call (or an interest rate swap where you received fixed and pay floating).
On Question 3, we are short a call and therefore we are long stock as a delta hedge. Let's consider two cases separately:
(a) The stock price falls, so the delta of the option position is reduced in absolute value (it is negative and moves towards zero), we have to sell some of our hedging stock to match this decrease.
(b) The volatility rises, what do you think happens to the delta? I think it moves towards -0.5, but without knowing if it is above or below this now, I don't think we can predict the direction of change, up or down.
If I answer Question 4, do I get the job?