Best-responding to a stochastically higher distribution of bids
Here is a sketch of an answer.

As noted in the question, the optimal $b$ must satisfy

$$b = v - \frac{F(b)}{f(b)}$$

Let us define the function $h(b) = v - F(b)/f(b)$. Our problem is to find the the fixed points of the function $h(b)$, i.e. the $b$ such that

$$b = h(b)$$

What can we say about $h(b)$? Assuming there is a unique optimal bid, it crosses the 45 degree line exactly once. Moreover, assuming that bids cannot be negative, i.e. $F(b) = 0$, it follows that $h(0) = v$. Thus, assuming that $v &gt; 0$, the function $h(b)$ starts off above the 45 degree line.

Now, when the opponent bids are 'stochastically higher', the function $h(b)$ moves upwards in the sense that the 'new' function lies everywhere above $h(b)$. It thus seems reasonably clear that the unique fixed point must increase, though admittedly I lack a conclusive proof of this fact.

I am also unsure about the bonus question (whether my new bids seem stochastically higher in the sense of the reverse hazard rate).