Lot of questions. Lets run through them one by one:

How do we have these letters at all?

Yes, these letters were kept and archived by the authors. Because these people are famous, there was a lot of interest in their library and correspondence after they died. So people wouldn't just throw it all out without looking at it. Instead, it would end up in collections, libraries and archives.

However, a lot of correspondence gets lost in history. For pretty much every important figure we have missing correspondence, and every few years someone will find a missing piece of correspondence in some archive or other. Take for instance the missing letter of Descartes, which was found all the way in the USA out of all places. The story can be found here: http://www.nytimes.com/2010/02/25/books/25descartes.html?scp=1&sq=descartes&st=cse

To give a personal anecdote: I once went to read some letters between Einstein and someone else (can't even remember who). The letters survived because they were kept by the family. Then, the family gave the letters to a Dutch archive. And now they are just kept there, available for anyone to go and read.

How were they sent and received?

There are a number of ways. Direct mailing was an option in the 17th century. But a lot of communication between scientists was mediated by important go-betweens. In the first half of the 17th century, for instance, Marin Mersenne was the pivotal figure for intellectual correspondence between pretty much all of europe's biggest minds.

Was it known that they would be read in route?

Sometimes people were concerned about this, yes. See, for a nice anecdote about this, you can watch this video: https://www.youtube.com/watch?v=6beq3aHqHps

• What form did these letters take? (Were they rolled up pieces of paper? Placed in envelopes of some sort?)
• How were letters formatted? I know some of Newton's letters had titles, for example. Was he just writing out papers and sending them as letters?
• What form did addresses consist of?

If you want to see what such letters looked like, you might like this collection of correspondence: https://archive.org/details/Correspondence00Cler I think it answers your questions a bit.

Where are these letters now, generally speaking, if they survive?

In archives across the world. The Royal Society keeps a lot of correspondence of its members. There are hundreds of other important archives across the world. It depends on who you do research on. For instance, Boyle's correspondence is kept at the Royal Society. You might find it interesting to read the introduction to the Boyle collection. It gives a lot of insight into how the correspondence survived: http://cisupa.proquest.com/ksc_assets/catalog/3469_BoylePapersLtrs.pdf

Sorry if I didn't really give a good response on some questions. It's all pretty difficult to explain in words. There were a lot of different ways of sending letters, and a great variety in how authors used them. Some were private, some public (e.g. the Leibniz-Clarke Correspondence). Some had titles and were very well thought out, others were just simple friendly correspondence.

But I guess the bottom line is that the mail system was pretty much what we would expect: you write a name and location on the outside of the mail, and give it to the post master. They make sure it arrives. The biggest difference would be how slow it was (a week to go from London to Edinburgh), the unreliability, and the costs involved.

Edit: also, there is a postal museum apparently. They have a lot of historical information too http://www.postalheritage.org.uk/explore/