0:17- The squiggles are a little bit random, but they may be text.
0:19- This is in a cave of some sort. The crystals growing in it do not resemble luminous ore deposits. Zelda and Link are using a water buffalo to carry their supplies, which are found in Lanayru, East Necluda, and... Faron. Most notably in Faron, which is where the Zonai people lived. The wall carvings seen in this shot resemble architecture built by the Zonai people. In the Faron Woods there are statues of boars (represeting power) that resemble the animal on the wall. In addition, that animal head is seen in the three labyrinths built by the Zonai.
0:22- First shot of Zelda and Link. Zelda has a haircut. Link has been redesigned slightly in regards to his wardrobe. He's wearing the Hylian Hood, pulled down (please let this be a feature in the game) and a Hylian Tunic with the cloth replaced by the Champion's Tunic. I'm a fan of this redesign both aesthetically and because it bothered me that the Champion's Tunic had the most defense yet no mail armor.
0:25- Zonai Tower in the bottom left corner. The red rock in this cavern is also seen in the Faron Woods and in the Gerudo Highlands.
0:36- THE HAND. I don't want to speculate too much about this since it seems to be super relevant to the plot so there isn't anything definitive to say.
0:40- This is Ganondorf.
0:41- Here we see the hand merge with Link's right hand. Here's a hot take: one of the ideas for Link in BotW's development was for him to not have a right hand. Instead, it would be bionic. He could change between weapons. Perhaps that will be a feature in this game?
0:41- Malice?
0:41- This shots are happening quick, but it shows Link and Zelda vertically joining hands. Is one helping the other up a ledge, or saving them from falling down?
0:42- This appears to be the entrance to either a dungeon or the cavern in the trailer. It greatly resembles the dungeon entrances from the original Legend of Zelda. It should also be noted that it is carved into red rock (again, like the cavern) and has some light white dusting around it, which makes me think that perhaps it could be in the Gerudo Highlands. I'm not sure how this lines up with the Hyrule Castle scene seen later in the trailer, which suggests that they are in or below the Laboratory.
0:42- Vertical hand joining again, but this time with THE HAND. THE HAND looks like one belonging to a Sheikah Monk.
0:43- THE HAND lessens its grip on Ganondorf. But why? Another plot point, apparently.
0:43- The ground is collapsing, which makes me think that they really are below Hyrule Castle somehow.
0:44- Ganondorf begins to move and comes back to life. Yikes.
0:54- Hyrule Castle launches into the sky. Please note that the towers are missing. They were present at the end of BotW. Look at Akkala Citadel for proof. The Divine Beasts are gone as well.
Speculations:
This game will delve into the backstory of the Zonai.
Ganondorf will play a larger role.
The game will not take place in Hyrule. This would be awfully boring as players have explored almost every square inch, and little would have changed. Furthermore, every sequel game has taken place somewhere other than the first (Termina in Majora's Mask, Holodrum/Labrynna/Koholit for the GB games, as well as different parts of the ocean for the DS entries).
Hot takes:
Link may have a bionic arm.
Play as Zelda.