Is there a set range that the rifle will only be accurate at? For instance, if you zero it to hit bullseye at 100yds, will a 200 or 300 yd target not hit exact center? How far off would it be?
When you zero your rifle (any firearm for that matter with any type of sighting system), the bullet will cross your line of sight (provided the sighting system is mounted directly above the bore and not offset to one side or the other) at two places (unless you zero at the apex of the trajectory of the bullet path). For instance, for the M16/AR15, at 25m and 300m is pretty close to where the bullet crosses the line of sight.
Due to the curvature of the path of bodies propelled through the air (with out additional on-board propulsion), and the ballistics of the .223 and 5.56, the bullet cannot be zeroed at 100yds and 200yds nad 300yds at the same time. You would need a laser for that if you are on Earth or any other planet that has gravity. (Even in outer space, the bullet would only cross the sight line once.)
This is the reason why the standard rear sights of all rifles have some type of adjustment to compensate for the different ranges. Could be a a sight that is on a pedestal that moves up and down like an AR, a ladder like on an AK, a paddle wheel like on a CETME, etc. It sounds as if your AR did not come with a rear sight and you may not have a reference to see the adjustments the regular AR rear sight can have as the Troy, IIRC, does not have any elevation adjustment other than the two apertures. Check out a standard rear sight of an AR.
I was thinking about it this way... if the sights are 2.6 inches above the bore, and you sight it in at 50 yds, does that mean at 100 yds the bullets will hit 2.6 inches below center?
Only if you are using a laser. Due to gravity, it will be less than 2.6"
above the line of sight. That is, the flight path of the .223/5.56 is still going up at 100 yds when zeroed at 50 yds.
(Not to scale.)
When the bullet leaves the muzzle it will rise into the line of sight of the rifle sights and continue to rise above the line of sight until gravity begins to pull the bullet down.
Gravity affects the bullet the instant it leaves the muzzle. The reason the bullet 'appears' to rise is that the rifle (or any firearm) is very slightly pointed upwards so that the flight path of the bullet will meet the line of sight. Their is no aerodynamic lift imparted by the bullet.