If you can fix the object weight, this will be simpler. If you can say "object will weigh 45 grams plus minus 3 grams" for example, then you don't need to include scales in the launcher.
If the object is ferromagnetic, you can build a railgun-like launcher. If not, you will need to either build a slingshot-like launcher with a spring of some sort, an actuator to pull it, and a release to release it, or you will need to do something pneumatic. Pneumatic might actually be simpler. You'll need to measure the kinds of force that can be produced, and calibrate your control appropriately.
Another kind of actuator that might work is one or more rubber wheels that give something an initial velocity; this is used by some auto-firing Nerf guns for example. It depends on the shape of the object. Or you might be able to use some kind of sled that mates properly with the actuator.
The pivoting is simple with two servos (heading and pitch.) Make sure to get reliable, strong servos with good control, such as Dynamixel AX-12 or MX-28. The math for how to change the firing force based on angle and distance is very simple trigonometry and calculus, which is easy to put into the coding microcontroller.
To build a system like this and make it robust, expect to spend a lot of time doing trial-and-error and go back to re-work things based on learnings. Also, if the cost of the final components is $X, expect to spend at least 3 * $X in various research/testing.