I'm considering the conversion of my flat-bar hybrid bike (GIANT Escape 3) in to a makeshift road bike, complete with drop bars. However, if the derailleurs have to go, it may not be cost-effective to do so.
The Escape comes fitted with Shimano Tourney derailleurs on front and back. They've got white logos on them which say "SIS Index", as seen in this photo of the front derailleur. I've been led to believe that Shimano drop-bar shifters will be compatible with these. Please corroborate?
The rear derailleur shifter needs to be Shimano compatible (and 7 speed) if you want to use indexing (as always, friction always works), while the front derailleur will need to be replaced with a Shimano road FD or you will need to use friction shifters (bar ends most likely) for the front.
That being said, the bigger problem is the bike fit with these conversions -- the geometry for drop bars is not the same geometry you use for a bike with flat bars for the same person. We've got some good answers on this here (particularly the one by sixtyfooterdude). You'll also need Problem Solvers Travel Agents or special brake levers for the linear pull brakes, so in all likelihood you'll spend significantly more than the bike is worth on doing this upgrade (which might not give you good fit) -- you're better off just buying a road bike (in fact, on such a cheap bike like the Escape 3, you're almost surely better off selling the bike and buying a road bike unless all the parts and labor are free and the result fits you).
You should not have any problem with regards to your rear derraileur. it will work with a wide variety of shifters. I've mixed many derraileurs within the Shimano brand across various product generations (usually from 7 speed to 10 speed). My limitation would be I've not used 11 speed equipment. There are some exceptions. For example the Shimano AX series derraileurs from the 80's will only work with friction shifters. .
For the fronts, there is some information on Sheldon that indicates that there may be issues with brifters and your derraileur. I've used brifters with a very similar derrailuer to to the Tourney and it works ok. My only concern would be the triple cranks that you have and its compatibility to a triple front shifter indexing. That I cannot comment on