A fast cricket-bowler or baseball pitcher does 100mph which exceeds the traffic speed limit . If his 2ft arm is just 6 times longer he is at the sound barrier ,  and double that is a rifle bullet Mach2  .  16 times that is escape velocity for earth orbit.   About 170 yds / 150m. long arm.

So you get the Greek army pulling a rope with gearing at 4mph which needs 25:1 to reach 100mph at the inner arm. Mach 33 at the end of the arm and release the Zeus javelin at climbing angle .

With regards to the first point- yes- IF you can maintain the same angular velocity. That's a big IF.

There's a lot of limiting factors, mainly anything with any respectable mass launched at mach anything (let alone mach 33) will put an insane level of stress on the components that are used to launch it. Just to accelerate the huge 150m lever to baseball pitcher level angular velocity would require tremendous torque and definitely destroy whatever lever or motor was used long before it gets there. You can make the components stronger but then you will require more force because they will be heavier, and so on.

Additionally, at high speed various frictions like air resistance become very significant. So power doesn't really scale to the speed achieved. For example, a bugatti now has 1500hp but it's definitely not 10x faster than a 150hp car, more like 2x faster.

And the correct altitude.    

So, would you go for a high lobbed shot where the projectile gets up to the stratosphere or a line drive shot with just enough arc and velocity to clear the mountains between them?

I like railguns. Any gun that fires a projectile so fast it sets the air on fire is fine by me.