1. In your own opinion, could Buddhism exist without any one of the following: monks, nuns, laymen, laywomen?
I would think that there are two different levels to this question. The first being that in order for Buddhism to exist with its rules, precepts, philosophies, and other various teachings – there needs to be some sort of guide along the way to keep it to its true roots. For example, I will use the analogy repair manual. There are people who are very handy, sort of handy, and no so handy when it comes to fixing things. With a book a person can have a anything from a very simple to a very complex system of repair work. And while a book or manual offers a guideline for understanding how something operates, it may be necessary to seek out someone who better understands how to use the book. Or even better, someone who knows better on the subject and has been in certain repair scenarios before. Some things may or may not be covered in the manual as well.
On the second level I think the idea of Buddhism in any sense for personal enlightenment or the addition of empathy towards others can exist without monks, nuns, laymen, and laywomen. What I mean by this is that even if Buddhism didn't exist, I think people would still try to better themselves and others. They would set their own guidelines and limits and may even write a manual along the way.
2. What are the advantages of going on a pilgrimage?
The advantages of going on a pilgrimage are to distance oneself from their current state and to reflect along the way. They might also go to a place where someone such as, the Buddha resided to feel more of a connection and find inspiration in themselves. I suppose in a sense a pilgrimage is about feeling a connection mostly but not entirely. Never having been on one myself, I would also imagine it might be a much more heightened experience for the closeness one could feel to the Buddha.
3. Are monks and nuns as reliant on laypeople today as they were in ancient India? Are laypeople of today as reliant on monks and nuns?
I would say that there is always going to be a symbiotic relationship between the two sides. However, I think that the reliance between the two has become less – more so in the sense of needing the basics to survive (i.e. food and shelter). This is mostly due in large part to the growth of industrialization, transportation, and technologies. We have more: planes, bridges, roads, cars, boats, computers, factories, houses, shelters, and the list of excess goes on - than ever conceived of during ancient times. On the teaching aspect alone, I would even say to some extent due to media access such as, the internet, even the teachings can be acquired without the need of an physical person. That being said however, I suppose if one was to boil this down to salt or the cause and effect concept then yes, without the teachings originally put in motion then the internet or other places of information would not be available. So yes, there is still a need but not as much as it used be.