Characters in a Lord Kroak + Temple Guard Unit?

1. Yes.

BRB p. 100: "Whilst a character is part of a unit, both he and the unit (including any other characters that have joined that unit) are treated as a single combined unit for all rules purposes, save for the exceptions listed here."

None of the exceptions listed are applicable. And while the Unbreakable special rule specifies (BRB p. 78) that "Characters that are not themselves Unbreakable are not permitted to join units that are (even if a character was to become temporarily Unbreakable for some reason, because of a spell or suchlike)," that is not the case here either, as you yourselve observe [edit: as long as the Oldblood joins first. Likewise, while the Oldblood cannot join Kroak, the inverse is not true: Kroak could join the Oldblood, but the new unit would not benefit from the Unbreakable special rule. Note that the same is true of flying units. A character cannot join flying units, but a flying character can join non-flying units.

Furthermore, the wording of Eternity Guardians special rule itself specifies "all models in the combined unit gain the Unbreakable special rule," which includes any other models in the unit, not just Kroak and the Temple Guard.]

2. Yes, although strictly speaking, you should first deploy the Oldblood and then Kroak.

On a side note: the BRB itself does not actually specify "deployment" as a way of characters "joining" units. It merely states that (BRB p. 97) "To join a unit a character must move into base contact with it during the Movement phase." However, it is the logical consequence of deployment in a unit, and BRB Official Update Version 1.9, p. 10 does say as much:

Q: If a character is required to join a unit and all units are required to test to see if they are held in reserve or moved on from a board edge at the start of a turn, how is this resolved? (p97)
A: Before rolling for any units you must nominate which unit each character that is required to join a unit is going to be deployed with and then roll once to determine if they all turn up or all are delayed.

3. If "no" had been the right answer, the answer would have been the first option. There is no rule allowing a character to leave a unit in this way, and there is no rule requiring Kroak to join the Temple Guard.

Not sure what you mean there. In both cases the character without the special rule cannot join the character with the special rule, but the character with the special rule can join the character without. Because of the Unbreakable special rule, an Oldblood cannot join Kroak, but Kroak can join the Oldblood; because of the Fly special rule, the Oldblood cannot join Tiktaq’to, but Tiktaq’to can join the oldblood.
In this regard, the main difference is that characters with the Unbreakable special rule can, in fact, join other models with the same special rule, while characters with the Fly special rule cannot(unless specifically stated otherwise).