I am building a gearbox that needs to lift around 50lbs very rapidly. I already have a single planetary gearbox that does the job in around 7-10 seconds, but I would like to cut this time down significantly. In order to do this, I have heard that I should be using a gearbox with multiple input motors as this will be stronger. However, I don't know how I would calculate the torque provided or the speed with which the output shaft will be turning.
The first order approximation, likely sufficient for your needs, would be to just sum all the "motor-side" factors and treat two motors and their axis-attached gears as a single motor and a single gear of twice the torque, twice the friction, twice the durability, twice the moment of inertia, same diameter, and just calculate it as for a single such motor.
It becomes trickier if the motors or their gears are different, because different speeds and torques don't add up that nicely, but if you just double the input, simple doubling of the parameters and treating it as a single input will cover most of of your bases.
Effect on durability of the output gear is changed, as two motors distribute the load, vs one gear of twice the torque being able to snap the teeth or cause skipping, but if the output gear can take input from the single, double-torque input, it will take it from two single-torque ones easy.