Shove = push ( in a sudden and rough manner, generally quickly )
Haul = pull ( something heavy and cumbersome, generally slowly)
Toss = throw in a light and often careless manner
Cast = only generally used for fishing nets and lines, or in certain fixed collocations such as 'cast light', 'cast doubt' or 'cast aspersions'. Not a synonym of 'throw' in other contexts.
Launch = used of boats, ships and rockets. In business, used for products, services and campaigns. Again, not a synonym of 'throw' in other contexts.
Here's some advice: when it comes to vocabulary like this, a translation dictionary is the worse friend that you can possibly have. There's no point in saying to yourself "All these words mean the same to me because they all translate as X in my language".
Throw/chuck/hurl/fling your dictionary away! [ Just a few alternative words for you there ;) ] Advanced learners of English need to accept that the English language has a rich and varied vocabulary, and there will inevitably be many different ways of expressing similar ideas. Forget about what you think that these words 'mean'. Don't ask for 'explanations'. Instead, look at examples of real English. Google yourself some real sentences and find out what contexts and collocations these words are used in. Explanations such as the ones above have limited value - instead, you need to expose yourself to some real world language.