How to prove that profit function is convex in price (with smaller price)?
I don't think it is a case. On a page you linked we have a proof that the profit function is non-decreasing in $p$. If the output price decreases, $p \geq p'$ and factor prices remain constant, $w_i \leq w'_i$ for all inputs, then the profit would be less or equal to the previous one. 

Then, in connection to your previous question: How to prove that a concave production imply that the input requirement sets are convex? you can let $\mathbf{y}=(y,-\mathbf{x})$ be maximizing profit at $(p,\mathbf{w})$ so that your profit function becomes $\pi(p,\mathbf{w})=py-\mathbf{wx}$. On page 12. of the Lectures in Microeconomic Theory Fall 2009, Part 3 you have linked, you can see a proof of that (probably the same as on the page you linked just now but not sure as the page does not open properly to me (cannot really see the mathematical equations).

As for your second question - "reverse convexity" is a little bit strange expression, to show what the author meant I will use a picture. 



As you can see it is still convex in input prices $w$. However, in comparison to plot with output prices $p$, it is decreasing.



Algebraically, you can see it that way:


Profit function is convex in both prices: $$\frac{\partial^2\pi}{\partial p^2}&gt;0 \;\;\; \textrm{and} \;\;\; \frac{\partial^2\pi}{\partial w^2}&gt;0$$
Profit function is increasing in output prices $p$: $$\frac{\partial\pi}{\partial p}&gt;0 $$
Profit function is decreasing in input prices $w$: $$\frac{\partial\pi}{\partial w}&lt;0 $$