a. He was very much annoyed.
b. He was annoyed much.
c. He was annoyed very much.
I think (b) and (c) sound a bit strange.
I think (b) and (c) could mean that he was often annoyed. I think could refer to a single incident as well.
d. He was annoyed much when he leaned that his son was arrested.
e. He was annoyed very much when he leaned that his son was arrested.
Is that correct?
No, it is not correct.
You cannot follow He was annoyed just with the word much although, strangely, you can say that He was much annoyed, meaning very annoyed.
To say:: He was annoyed much. leaves the reader wondering much what?
But much can follow He was annoyed when it modifies something else. For example:
He was annoyed much more often than I expected.
Your examples d. and e. are constructed in similar vein.
Both are legitimate although d. definitely reads more easily when much precedes annoyed. Most English speakers would probably reverse the word order in e. as well. But that's not to say that it's wrong, just a little unusual.
Having said that, it's possibly to use the adverb muchly before or after annoyed although, as most dictionaries point out, it's generally in informal or humorous contexts.
