Suggested answer:
(1) In the corpus angry typically precedes the prepositions about, at, and with, e.g.:
He was angry about class...
Soon he begins to get angry at the people involved...
...that made him angry with Philip.

In the corpus annoyed typically precedes the prepositions about, at, by, and with, e.g.:
My mother was very annoyed about it at the time.
...she was annoyed at Bill's attitude ...
He was proud of her and annoyed by her.
She was now beginning to feel annoyed with them for being so late...

In the corpus furious typically precedes the prepositions about, at, and with, e.g.:
I was furious about the fate of my pound note.
Lee Ann was furious at me.
Clarence was furious with her.

(2) While angry about typically refers to something you are angry about, angry at and angry with typically refers to someone you are angry at or with. Although there are counterexamples to such a generalisation, these are tendencies shown in the corpus.

(3) As regards the adjectives + about, the tendencies are similar to those of angry about. Although few, the occurrences show that annoyed about and furious about generally points to inanimate things.

With at, annoyed and furious seem to differ from angry in that they refer to something rather than someone in the great majority of the occurrences. But again there are few instances to base this on.

Finally, annoyed with and furious with tend to refer to persons rather than things, i.e. the same as for angry with, although there are counterexamples.

(4) Yes, the two adjectives are found in combination with the same set of prepostitions as the "negative" adjectives, i.e. about, at, by (only found in combination with annoyed above), and with:
His parents were absolutely delighted about it.
...was clearly delighted at the compliment.
Courbet was delighted by this idea...
He was delighted with his room.
Okonkwo was inwardly pleased at his son's development...
I wasn't too pleased about that.
I was pleased by this.
Dickie was very pleased with the interview.

Read about adjective + preposition combinations here and here
Terminology: preposition