Learn English – What’s the meaning of “out” here

You can nod at a person to acknowledge their presence. Here it has that same meaning, but figuratively rather than an actual physical nod.

You will see that the next sentence in the article talks about the new building having the marble from the same quarry as the original. So the "distinctive nods" mean that by using the same marble they are acknowledging the original World Trade Centre complex.

The following explanation from Wikipedia should give you a better description of how nodding can be used in this way. It talks about the physical gesture - the metaphorical one mentioned in your article would have the same sentiment behind it. The "degree of respect" part is notable - they are not just acknowledging the previous building was there, but the "nod" is a gesture of respect to it and what it represented.

Nodding can also be used as a form of nonverbal greeting or acknowledgement of another's presence; in this context, it is essentially an especially mild form of bowing, with just enough movement to show a degree of respect without additional formality. This includes the traditional downwards nod, or the upwards nod (which is more informal and usually used among friends or subordinates). To increase the formality, the downwards nod may also be accompanied by a suitable verbal greeting.

(from Wikipedia)

Literally it refers to "checking out of a hotel." It's something you do when you leave the hotel. Once you've checked out, you're gone.

This can be used figuratively in a few ways. The basic metaphor is that once you've checked out, you're gone. It could mean several things:

1. Death. It's what you do when you leave this mortal life behind. Once you've checked out, you're gone.

2. Senility. If you're still here physically but mentally you've checked out, then your body is still here but your mind is gone.

In your example, it's like #2 but perhaps without the sense of permanence. Physically speaking, you're sitting in class or you're at a meeting, but mentally speaking you're not there at all. It doesn't appear to suggest actual senility in this case, though—it's just being used to describe a lack of attention. This seems to me like a slightly more extended use of the metaphor.