Why? Because a ship will spend days building velocity during cruise, then have minutes to hours to alter it during combat.
I'm not sure I see your point. You seem to imply that velocity building
must take days. I would say this would be the case for freighters and early game ships, but warships would have high G engines (with some form of grav/inertial compensator, which would probably be the main feature of TN technology). Such ships would quickly accelerate to cruise speed (fuel cost being the same) and operate, most of the time, in ballistic mode, ie with engines stopped.
Even if high G maneuvers were too risky to be attempted out of battle, you could imagine a high G drive giving a low acceleration. Derivating Tsiolkovski's eq, you get the formula for accelerations (G), as
dV/dt = -v_exhaust/m dm/dt
To reduce acceleration, you could either reduce v_exhaust (which might not be practical in chemical rockets, but is perfectly doable if exhaust are accelerated by a field, as in a ion engine), or reduce mass output. In practice, it just means warships would have much more powerful engines than commercial ships, and probably some kind of anti-grav contraption to allow their crew and frame to sustain the high G delivered by the engine.
For all drives, there would be some speed over which maneuverability drops (because the engine cannot support such accelerations). This max-maneuver velocity would be low for old generation ships, or later day freighters, higher for military vessels, and much higher for fighters (because of the low m in the above eq).
A missile closing at .5c may be a difficult intercept, especially if set to detonate at long range with lase rods.
But those missiles would move in a straight line, bearing on your ship, no? So, if you can align with them (easy task if you are at rest, almost impossible if you are moving fast, but then, the missiles will have a hard time intercepting you...) and if your antimissiles have an area effect (just like enemy missiles), the only thing you need is that your AM "crosses" the enemy missile before it is in range of your ship.
Supposing your AM has the same speed as the missile (but you could adapt the formula), this means you can detect the enemy projectiles at twice their explosion range. But all this only seems to work if you can align your fire with enemy missile path, ie if you are very close to te ship targetted. "sideways" shoots at missiles, on the other hand, would be little more than potshots...
Francois