Shove has the same meaning as the word "push".
Example: "I was in a rush, so I shoved people out of my way to get to the front of the line."
Toss and throw are interchangeable. They have the same meaning. Toss is generally used if you want someone to throw an object towards you lightly.
Example: "Toss me the basketball, I can make the shot!"
Example: "That is a huge rock! Let's see how far we can throw it!"
Cast is not used as frequently as others. It is generally used while fishing. It is this action: https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwjj8KHQj5bdAhUCRqwKHUtBCMMQjRx6BAgBEAU&url=http%3A%2F%2Fstevenojai.tripod.com%2Fparacast.htm&psig=AOvVaw26FrDMYQj4W2SVB_jFMV8r&ust=1535764677153214
Example: When fishing, cast your line and wait for a fish to bite."
Haul has the same meaning as pull. Often haul is used when pulling a heavy object.
"After casting our fishing line, we hauled in a big fish."
Launch is similar to throw and toss. It generally is used to indicate a powerful throw. It is also used to describe a rocket ship takeoff: https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwiEvp_tlJbdAhVJ-6wKHRH4CvAQjRx6BAgBEAU&url=https%3A%2F%2Fgiphy.com%2Fexplore%2Fspace-shuttle&psig=AOvVaw1Wg9Uiyr_jFAqoX7c3VHCg&ust=1535766138086867
Overall, toss, throw and launch can all be used to describe making an object in your hand fly through the air. Toss is softer and launch is harder. Cast is unique and is used to describe certain things that are thrown, like fishing nets or lines. And haul is a verb that is similar to pulling, but usually used when you are pulling something heavy.
Hope this helps!