I have a Trek FX7700 that I would like to replace the flat bar with a rod (drop) bar. The bike has V-brakes with separate brake levers and trigger shifters. Can these be slid around and mounted on the top of a road bar in the similar position they are mounted on the flat bar? I would also install “Stoker Levers” for more hand positions. Any Pros/Cons to doing this?

I have the shifters on the round part of the bars. Did a similar conversion that you plan, but I rather ride on the drops, shift and brake. I have rather big hands and i can reach the brakes and shifters from the top as well ... with my fingertips.

Slide, wedge, massage, convince, curse, and they will fit

Better yet, buy a set of 105 STI levers and a couple Travel Agents for the brakes and do it up right. I have the same bike and did it that way.

Travel agents, Bleah never could get those stupid things to pull enough cable.
Went back to good old Cantis and some bar end shifters. I know the OP is trying to save
money, but hanging mtb hardware on a road bike is an annoying and temporary setup.
Bar end shifters and cantis are bomb proof. But that's just my opinion.

Good luck.

-Marcus

Or you could go with the Dia-compe 287V road levers that pull enough cable to operate V brakes. Lots of power and modulation. Forget travel agents.

Go with barend shifters, they're awesome. You could also mount barends on aero bars.

You're not installing them correctly. They work perfectly.

You can keep your flat-bar controls if you use a treking style butterfly bar.
Or you could fit some aerobars to your flat bars.