Lever linked to hatch and bridge which would open the hatch and bridge with the pull of the lever.
Hmm, bridges either raise or retract when receiving an 'on' signal. To start/stop a cart on a single lever flip, the easiest way would be to set it on an active roller pushing against a lever-operated door. When the door is closed, the cart stays in place, once the door opens, the cart goes through. If you want extra safety against building destroyers, use a raising bridge instead of a door to hold back the cart. It will take longer to react to a lever pull, but can't be smashed by angry trolls (although a troll smashing your door and getting a face full of minecart for its trouble isn't exactly unwelcome).
1. How many rollers would I need to get the cart up to killing speed?
Roller length is largely irrelevant, to accelerate a cart to the roller's nominal speed, a single tile is enough. "Highest" speed is already the default setting. As long as you use heavy carts, the effect on not-gargantuan creatures should be decent. A heavyweight cart often sends smaller/lighter opponents flying, which can cause all kinds of nasty damage, but with roller-accelerated carts, you usually need a bunch of hits to wear down a foe.
2. What happens when it hits the 3 trackless, but otherwise flat, road tiles?
It passes in a straight line. As far as i know, possible passengers or cargo should stay in the cart. The cart will experience more deceleration on non-track tiles, but the effect will be comparatively small when looking at a mere three tiles.
3. Would my dwarves automatically haul the released minecart back into position? Or would I have to manually rebuild it?
As soon as a minecart stops, dwarfs will go and drag it to an appropriate route stop. They will not check if that location is actually safe for placing a minecart - they might place it on an active roller or throw it into the hole under an open hatch cover.
4. How many tiles past the road would the cart remain a deadly projectile?
5. Would having/not having those track tiles on the opposite side of the road impact its effectiveness as a weapon?
With a max speed roller, friction isn't a big issue. The cart could go over 1000 tiles (a complete embark map is about 200 long/wide) in a straight line along track, about 60 tiles over non-track floor.
And i agree with jcochran - build it and try it out. Building things is always much more fun (or FUN, or !!fun!!) than just talking about them.