Clarifications

The advantage/disadvantage happens when a creature can't be seen - being invisible is one way of doing this but it is not the only way - being hidden or in the dark or having a blinded opponent are other ways. See What advantages does hiding have? Because of this I will use "unseen" when I mean that the creature cannot be seen for any reason and "invisible" when I am talking about having that particular state.

You have used the word "(opportunity)" in a way which makes me think that you are coming from an older version of D&D where casting a spell could trigger an opportunity attack; this does not happen in 5e! The only trigger for an opportunity attack using a reaction is if a seen creature voluntarily moves out of another creatures reach without taking the Disengage action (PHB p. 195).

Answers

Two things to keep in mind:

1. The sequence of events goes: attack or cast spells then lose invisibility.
2. Each attack occurs at its own time - there are no simultaneous events.

If you keep that in mind then all becomes clear the creature :

1. gets advantage if it is unseen when it attacks, and
2. causes disadvantage if it is unseen when it is attacked.

Question 1

1. The creature gets advantage when it makes an attack while unseen - it then loses invisibility.
2. when the creature is attacked disadvantage happens if it is unseen at the time of the attack - this does not cause it to lose invisibility.
3. See 1. above. If a creature then attacks again it is no longer invisible (it could still be unseen e.g. a creature with darkvision attacking one without in the dark); so see 1. above.

There are all sorts of sequences that can affect this, for example:

• you are invisible at the start of your turn and can attack twice when using the full attack action and can cast invisibility as a bonus action. I am not aware of any creature that can do this but its theoretically possible.
• You move 10 feet,
• Take the Attack action and attack with advantage because you are unseen,
• turn visible,
• you use your bonus action to turn invisible,
• move 10 feet to another opponent; you do not trigger an Opportunity attack because you are unseen,
• attack then with advantage because you are unseen,
• that opponent has taken the Ready action to "attack the first creature that attacked them", they attack with no advantage or disadvantage as you are now seen.

Question 2

In D&D 5e things do exactly what they say the do; no more, no less. Just because an ability and a spell cause the same effect does not mean they have anything to do with one another beyond that.

Superior Invisibility. As a bonus action, the dragon can magically turn invisible until its concentration ends (as if concentrating on a spell).

This does not refer in any way to the spell Invisibility! The limitation on attacking and spell casting applies to the spell; it does not apply to this ability. This ability ends when the faerie dragon stops concentrating - they can otherwise attack and cast spells and remain invisible.

I can find no Sage Advice regarding the matter, but the rules about unseen attackers starts with this:

Combatants often try to escape their foes' notice by hiding, casting the invisibility spell ...(PHB 195)

This seems to suggest what I believe every table uses: If you are invisible, you attack someone who cannot see you with advantage, even if doing so breaks the invisibility.

Regarding nocking an arrow or swinging back for a punch: this would be a Ready action and not Attack or Cast a Spell, and thus would not end the spell invisibility.

If you cast a spell with a longer casting time

you must spend your action each turn casting the spell (PHB 202)

thus you reveal yourself on the first turn as you have taken a Cast a Spell action.