What would be the cheapest way to get video signal

If you're just trying to overpower a video signal without gating it, your effective source impedance is going to vary every time your video is switched on and off. Not good. You need to ensure that both sides of your video connection see a uniform 75-ohm impedance whether you are passing through the signal or supplying your own. The easiest way to do that would be to use a pair of video amplifiers and put your circuit in the middle. The input amplifier should probably have 1:1 gain and a low-impedance output; use something like a 4066 to switch that and a white signal. The output amplifier should have a 2:1 gain and a 75 ohm resistor in series. BTW, are you doing anything for DC restoration? That could be an issue as well.

One important factor when working with stepper motors is hold torque. You can do some simple calculations based on your camera weight and platform radius to determine the torque due to gravity, and your hold torque should be a fair about larger than that or the motor will slip. You'll also need stepper motor drivers in order to control the motors and provide enough current as well as the proper signal to control the motor. I've had experience with the Pololu A4988 I'm conjunction with Arduino; it's simple to use and only used 2 digital IO pins (at minimum). You can also spare more pins and control things like the driver enable, etc. As for actually choosing a motor, that can be tough. I'd pick a price range and form factor that you'd like to be within and use that to narrow it down. I'd also stay away from the spark fun motors as they are usually on the more expensive end and not the best bang for the buck.