It’s practically impossible (-20 penalty) to hide while attacking, running or charging.

So you can remain hidden while making a spring attack, but the DC to spot you is not going to be very high. If you still manage to win the opposed roll, your target will be denied its Dexterity bonus to AC and will have an additional -2 penalty because you are effectively invisible.

From Skip Williams' column:

If a foe you cannot see hits you with a melee attack and is adjacent to you at the time, you know the foe's location. For this reason, smart foes move right after they attack; even a foe that has made a full attack can move after attacking by taking 5-foot step (provided it has not already moved during its turn).

When an unseen foe hits you with a melee attack from more than 5 feet away, you know the general direction from which the attack came and that the attack came from more than 5 feet away, but you do not know the attacker's location.

That is the main reason to spring attack: you need to move after attacking, even if invisible, otherwise your target will know your location. If your target has failed its spot check, you still have total concealment (50% miss chance).

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.