An adverbial modifier is an adverb that modifies some other word. Wikipedia notes this distinction between modifiers and non-modifiers:
For example, in the English sentence This is a red ball, the adjective red is a modifier, modifying the noun ball. Removal of the modifier would leave This is a ball, which is grammatically correct and equivalent in structure to the original sentence.
[...] In His face became red, the word red might be called a complement or argument of became, rather than a modifier or adjunct, since it cannot be omitted from the sentence.
You can read more about adverbials at their own Wikipedia article.
This noted difference is quick to check so let's look at the examples you noted:
1) He worked his company hard. / He worked his company.
"Hard" is an adverbial modifier that modifies "worked".
2) They rehearsed every morning from ten till two. / They rehearsed every morning.
"Every morning from ten till two" is an adverbial modifier that modifies "rehearsed".
5) If they played a scene well he cried like a child. / If they played a scene he cried like a child.
"Well" is an adverbial modifier that modifies "played a scene".
6) He would skip about the stage on one leg. / He would skip about the stage.
"On one leg" is an adverbial modifier that modifies "skip about the stage".
In the future, if you are concerned about homework questions I recommend asking your teacher or professor for help as they are typically the final authority on which answer is correct. And they are, presumably, there to teach you this material.