You really have two questions there. I'm going to answer this one:
In a section like the chorus, how do you get the layering of many instrument to get that full and rich impression?
Playing a bunch of instruments at the same time can easily lead to a big mess, even when the instruments are playing compatible notes. Playing the exact same thing on the same kind of instrument at the same time (doubling) can actually sound smaller and thinner. Here's a bulleted list (my favorite kind of answer, I guess) of concepts on how to arrange instruments to sound good together:
When layering instruments, they actually need to have some separation in order to not fight with each other. Instruments can be separated by frequency, time, and "space".
- Different instruments normally have different prominent frequencies. Being aware of the different frequencies prominent in different instruments and taking advantage of that is the most effective way to layer instruments. For example, arranging a bass guitar to play the lowest notes, a guitar to play the middle notes, and a piano to play the highest notes is a good way to have all three instruments playing at the same time without fighting each other.
- The more instruments that you want to fit into a mix at the same time, the less frequency range each instrument needs to have. For example, if you are in a band playing guitar and there's a bass player in the band, you don't want your guitar to have too much low end, because that will fight with the bass. Also, the bass player should be careful about having too much upper midrange in their bass tone, since that will fight with the guitar. Each person should stake out their frequency territory and stay inside it.
- You can get the fullest sound by making sure you have one or more instruments covering the entire frequency spectrum and not leaving any gaps. This is a challenge because gaps will lead to a less full sound, but overlap ("fighting") will lead to a less clear sound. Along with timing, this is one important thing that makes the best bands sound so good, they know how to fill the spectrum with a minimum of overlap. The same goes for orchestrations, arrangements, and compositions in general.
- You can also separate instruments in time. Of course instruments can be played at different times to provide separation, and the timeline is a bit like the frequency spectrum: you might want to fill it up without gaps or overlap. But there are other time related aspects of instruments that can help provide separation.
- Instruments each have their own volume envelopes. That is, they each have a pattern of how the loudness of the instrument changes over time. Drums have a very loud initial sound and then they very quickly become much quieter. A B3 organ or a synthesizer or a violin might have the same volume the entire time it is playing, or a "swell" effect or technique might be used to make an instrument start off quietly and then get much louder. By mixing and matching envelopes along a timeline, you can get more instruments to play together in interesting ways even with overlapping frequencies. The most common example of this is drums. Drums have a fairly broad frequency response, so they would normally compete with every other instrument, but since the drum sound is so intense and so brief, it works to have the drums play at the same time as other instruments since the drums "win the fight" very briefly, and then go away leaving the other instruments layered as usual. Note that an overly busy drum pattern would usually stomp all over everything else.
- Another common use of envelopes in layering is mixing instruments that decay with more "pad" type sounds. Guitars, basses, and pianos start loud and then decay (much more slowly than drums do). Playing one or more of those instruments with a sound that stays the same volume, like organ, synth, strings, etc. (sometimes called "pads"), creates a layering effect where the decaying sound "wins the fight" at first and then as the sound decays, the pad "wins" out. This is a good way to keep the time axis full of sound.
- Finally, there is space (the final frontier). Our ears and brains actually have sophisticated localization abilities, meaning we can hear it when different sounds come from different places. With live, acoustic music, this helps a lot with layering. Watching a symphony orchestra (massive layering of sounds) live is very powerful partly because our ears can hear how each instrument comes from a different place in the room. When using a PA or in a recording, it's possible for the space aspect to become "collapsed" so that all the sounds seem to be coming from the same place. In these situations, we have to create space artificially.
- As Remco writes, the two primary space axes available for the majority of situations are left-to-right and front-to-back. Left/right space is controlled using panning and front/back space is controlled with reverb and delay.
- Left/right separation is pretty obvious. Changing the panning of a sound so that more of it comes from the left side or the right side (or the middle) helps our ears separate that sound from sounds that are not panned to the same position. You might have guessed by now that the fullest sound is created by having sounds coming from the far left to the far right with everything in between filled up with the minimum of overlap. Again, it's easier to understand that concept than it is to make it actually happen.
- Reverb and depth can be trickier. It might seem at first that putting more reverb on a sound makes it sound farther away, and putting less makes it sound closer. In practice it doesn't really work exactly that way. A sound can have a lot of long-lasting reverb and still sound very up-front. The secret is pre-delay. That's the time between when we hear the sound and when we hear the reverb of the sound. A longer pre-delay with reverb makes a source sound closer, and a shorter pre-delay makes it sound farther away.
- This might seem counter-intuitive but it makes sense if you think about how long it takes for a sound to get to your ears versus how long it takes to bounce off the "back wall" of the virtual room. A source really close to the back wall and far from you (the listener) will make a sound that bounces off the back wall almost instantly and so the initial sound and the reverb sound will reach your ears at the same time. A source right in front of you with both of you far away from the back wall will cause the direct sound to hit your ears almost instantly followed by a noticable delay before the first echoes from the back wall reach your ears.
What do I mean by a sound "winning the fight"? Really, most humans brains can only focus on one sound at a time. Normally, this is the loudest sound (not always the one with the highest volume - loudness is partly frequency dependent). In music with vocals, we usually want the vocals to win. When there are no vocals, the winning sound will be percieved as a melody or rhythm and as the "main" part of the piece. Controlling which sound "wins" means choosing what listeners focus on, and it's very important for making the most effective arrangements.
Also note that while the above tries to explain how to "fill up" the soundscape, as asked, it's usually better to only fill up the entire soundscape for part of a song or a piece. As mentioned in the question, it is popular and effective for there to be a contrast between the chorus and verse, for example, and one great way to make that contrast is for the chorus to have a fuller sound than the verse (or vice-versa).