I feel that enough time has passed since I posted my questions, so I am happy to announce
@TeaLovingDave as the winner of this set. Here are the answers:
1. Nankeen Kestrel (
Falco cenchroides). In at least the mid-to-late 20th century, the species was a quite-regular vagrant to New Zealand and it was hoped that the species would eventually start breeding there. The Nankeen Kestrel has never been recorded breeding in New Zealand to date, and has been seen in New Zealand much less frequently over the past twenty years.
2. Large-billed Crow (
Corvus macrorhynchos). The Yaeyama Jungle Crow (
C.
m.
osai) may be a full species.
3. Four: North America, South America, Africa, and Asia.
4. Velvet Scoter (
Melanitta fusca). According to the Handbook of the Birds of Armenia (Adamian & Klem, 1999), "The disappearance of this species as a breeding bird is attributed to the lowering of Lake Sevan, destruction of the main nesting area near the town of Sevan, and the conversion of Sevan Island into a resort peninsula". The Velvet Scoter still occurs in Armenia as a rare summer migrant or vagrant.
5. King Vulture (
Sarcoramphus papa).
6. White-eyed Duck (
Aythya australis) and Red-necked Avocet (
Recurvirostra novaehollandiae). The White-eyed Duck may have bred in New Zealand during the period when it was recorded there many times and it had a Māori name; Karakahia. The Red-necked Avocet is the rarer of the two species in New Zealand, with no confirmed records since a single bird that was at Westport from 1968 to 1970, versus the White-eyed Duck, which has had multiple confirmed records from the 21st century.
7. Ancient Murrelet (
Synthliboramphus antiquus). Its Japanese name in Kanji is 海雀; Umisuzume.
8. Rufous-necked Wood Rail (
Aramides axillaris).
9. At least eight species: Black-footed Albatross (
Phoebastria nigripes), Macquarie Island Shag (
Leucocarbo purpurascens), Grey Heron (
Ardea cinerea), White-bellied Sea Eagle (
Haliaeetus leucogaster), Corncrake (
Crex crex), Australian Crake (
Porzana fluminea), Common Moorhen (
Gallinula chloropus), and Red Wattlebird (
Anthochaera carunculata).
10. Rook (
Corvus frugilegus).