Shimano 3x7 rapid fire shifters

Your hybrid actually uses Shimano MTB cranks, freewheel, derailleurs and shifters. Any Shimano 3x7 shifter set will work with your current drivetrain.

OBC is right about compatibility, with the small disclaimer that Shimano road and Shimano MTB use different actuation ratios for the front shifter/derailer. But as you're staying within the MTB family it's all good.
Got a bit worried first when you said you wanted to move the drops inboard, I've seen a couple of really strange configurations over the years. But it looks like you're staying with a fairly sensible width for a stock drop bar, so no problem there.
Cutting the excess off, and putting some padding on the inside bar would give you something that's fairly close to a stock drop bar in both shape and functionality, so I'd probably go with that.
OTOH there's something to be said about the extra grunt and control provided by a wider bar, but only you can tell if that'd be a useful feature for you or not.
But if you intend to keep the excess for awhile, get the ends plugged before you hit the road. You'd be amazed at how bad you can gouge yourself, or someone else, if you crash with an open ended bar.

Some questions:
- Do the brake levers clamp well to the drop bar ends? I believe the drop bar ends has the standard MTB bar size of 22.something mm, while regular drop bar size is somewhat bigger.
- You're aware that regular drop bar brake levers has another cable pull than what's recommended for your v-brakes? Unless you've got one of the fairly few designed to work with V-brakes you will meet some issues.
- Have you checked the brake wire seat in your new levers? Flat bar brakes tend to use the barrel shaped wire stop while drop bar levers tend to use the button shaped one. You may not be able to reuse the old cabling.

- You mention "foam". There are foam tubes that can be used instead of bar tape, but it sure looks like you have "aero" brake levers. Aero levers are intended to have the cable run underneath the bar tape. Leaving it exposed makes riding on the tops quite uncomfortable. Don't know if it's possible to slide foam tubes over brake cabling.
- What about reach? Going to drops, particularly to the often used position of riding on the hoods, will put your hands something like 3-4" further ahead than before. More if you didn't use bar ends earlier. That's a big change in position. I find it a bit difficult to see how one could go between two such extremes and be content with both fits.