A Quiz Between Worlds

Answer: Lorule

Lorule is brand new to the series, acting as a complete counterpart to Hyrule. It has its own goddesses, Triforce, and even its own relative Link and Zelda equivalents. The reason that the two kingdoms are brought together is due to Princess Hilda's aspirations to seize Hyrule's Triforce, as Lorule's Triforce was destroyed.

The Dark Realm is the world we entered in "A Link to the Past" (SNES), which took place in the same Hyrule map as "A Link Between Worlds."
Subrosia is the world we entered in "Oracle of Seasons" (Gameboy Advance), beneath the land of Holodrum.
Labrynna is the land where our adventure took place in "Oracle of Ages" (Gameboy Advance).

Answer: Donate 3,000 rupees to a certain fairy fountain

This fairy fountain is located in a cave in Lorule. You must first pay 200 rupees to the bomb flower shop owner, who will then let you take a giant bomb any time you like. Then you take a giant bomb and walk to the cave just west of the shop, and detonate the bomb to blast the fairy fountain open.

Answer: Mother Maiamai

If you've found enough lost Maiamais and return them to their mother, she will be obliged to eat almost any item of your choice (provided that you own the item), and power it up. The Nice Bow shoots three arrows at once like a scattershot, reducing the need for precision aiming!

Incidentally, there is a fairy at the Pond of Happiness in a "A Link to the Past" (SNES) that can upgrade the amount of arrows you can hold in your quiver, after you donate a considerable amount to her pond.

Answer: Dark Palace

Gulley is indeed one of the seven sages that Yuga turned into portraits. Gulley's portrait is kept in Lorule's Dark Palace, or the counterpart to Hyrule's Eastern Palace.

Hyrule Castle doesn't contain any sages, but Lorule Castle does contain Princess Zelda.
The Skull Woods is where Seres' portrait is held. Seres is a sage, and she is the daughter of the priest from the Sanctuary.
Kakariko Village doesn't contain any sages either, but Osfala's portrait is held in Lorule's counterpart in the Thieves Hideout.

Answer: No

Due to the ability to rent almost every item from Ravio very early in the game, Link can access dungeons in any order. However, their completion can be easier if done in a certain order. For example, attempting the Ice Ruins prior to most of the others will leave Link in relatively higher peril, due to the Ice Ruins' more dangerous enemies, hazards, and more challenging boss battle. (I unintentionally did this, with much frustration.)

Answer: Bow of Light

Princess Zelda gives Link the Bow of Light, which can only be used when Link becomes a painting on the wall in the final battle against Yuga. Unfortunately, we are never given an option to save the game after the battle, thus the Bow of Light is only temporarily added to our inventory list.

The Red Mail is the last item that we can obtain in Lorule Castle.
The Tornado Rod can be rented and purchased from Ravio very early in the game.
The Fierce Deity's Mask is the last mask that Link can obtain inside the moon in "Majora's Mask" (N64).

Answer: Upgrade from the Net to the Super Net

According to the Devilish Girl at the front of the tower, no one has ever completed the Advanced level. Link does it twice, and each time you can earn a substantial amount of rupees as well! The Beginner level yields a prize of rupees. Completing the Intermediate level yields a Piece of Heart.

Answer: Majora's Mask

Oddly enough, Link possesses Majora's Mask in "A Link Between Worlds." This is most likely an easter egg from the game developers, as "A Link Between Worlds" is supposed to take place near the end of the Fallen Hero timeline in the series' canon. However, "Majora's Mask" (N64) takes place near the end of the Child Link timeline canon, and the two are inherently incompatible with each other.

Answer: Tornado Rod

Link needs the Tornado Rod to complete the House of Gales.

The Hylian Shield is the prize of the Turtle Rock dungeon and it can block magic attacks from Wizzrobes, but you don't have to have it to survive or beat the game.
The Net (or Super Net) can be used to damage enemies, or to capture bees, rupees, or hearts. Link can complete the game without it as well.
Finally, the Hint Glasses are a neat perk if you happen to get stuck in a place where the game designers thought you may likely get stuck. However, if you're attuned to "Zelda," you probably will be able to decode whatever obstacles the game offers without ever having to resort to the Hint Glasses, which can reveal a Ghini that may offer some advice.

There are some other items that are also unnecessary in the game!
Using the Foul Fruit damages all surrounding enemies if used, but the game can be completed without ever having to use a Foul Fruit.
In case of emergency, Link can use the Scoot Fruit to be transported to the entrance of the dungeon he's currently trying to complete. If well-prepared, Link will never need to escape the dungeon using the Scoot Fruit. It serves a similar purpose to Farore's Wind from "Ocarina of Time" (N64), except that the Scoot Fruit doesn't return Link to the room from whence he escaped.
The Monster Horn, the Monster Tail, and the Monster Guts are also not required items. They serve as ingredients that the Witch needs to brew various potions (east of Hyrule Castle), but Link doesn't absolutely need any of those potions to defeat Yuga.

Answer: A Link to the Past (SNES)

Indeed, "A Link Between Worlds" takes place in the same Hyrule map as "A Link to the Past." This was done intentionally, as the events in "A Link Between Worlds" are supposed to unfold much later in the same place, in the same "Legend of Zelda" canon.

Further, there are many other small details in the game that are reminiscent of "A Link to the Past," including Moldorm being the boss of the Tower of Hera, Grinexx being the boss of Turtle Rock (as opposed to Trinexx), Dharkstare being the boss of Ice Ruins (as opposed to Kholdstare in the Ice Palace), etc. The bird statues that Link activates also make a familiar tune when you save the game, reminiscent of the tune that would play when Link would call for the cucco to transport him in "A Link to the Past."
Also, Flute Boy from the bar in Kakariko Village is reminiscent of the flute boy's father from "A Link to the Past." Incidentally, Flute Boy is also more deeply reminiscent of some things from other "Legend of Zelda" games.