Unsorted mail (same goes for the waste) is produced by buildings similarly to "regular" factories. You can check it out by switching to the Cargo Layer view (or faster: by selecting a vehicle station, e.g. a bus stop, to be placed down somewhere). You'll notice that most of the buildings produce unsorted mail by converting it from COM, IND or sorted mail. Supply the required products and you'll get the unsorted mail.

Mail is generated in active, consuming towns. The more goods you deliver to a town, the more mail and waste the town will produce. Verify that the entire town is covered by caption areas of cargo bays, and collect the mail from all cargo bays. It starts slowly, but when all towns on the map are connected into the mail network, the amount of mail will become substantial.

During the start, I also use a few mail cars in existing trains, Later dedicated mail trains become interesting.

Simple truck stops do not collect unsorted mail. You need the cargo bays to do this. ;)

So will mail load at a passenger stop onto a assembler train? Like do I need to add a box car to my passenger trains to move it around? Are there any passenger cars that will take mail like in real life?

Mail is considered as freight, and not as a passenger. You need to handle mail via cargo stations. It should be possible to mix passenger and mail (box) cars in one train, but you will have to stop at a passenger station AND a freight station. But combining passengers and mail in one train is not practical because mail goes to, and comes from, the Postal Sorting Centre. There are (almost) no passengers that are interested to go to this unit.

So, mail is freight, and needs to be collected by cargo bays, and dropped of at cargo halts or cargo bays. All unsorted mail goes to the Postal Sorting Centre, and all mail from the Postal sorting centre goes back to the houses... all by freight vehicles. I hope this helps. ;)

Does sorted mail has a destination address in a specific city or is it just an unspecific product that can be delivered to any house asking for mail?
Is the idea that we should build a sorting center next to 3 or 4 towns and then connect all sorting centers or is mail just a local service that doesn't need connections to others?

Sorted mail has, the moment it is created, a specific destination. The destination will be chosen from any sorted mail receiver (house, HQ,...) to which there is currently a route possible. The unit of mail waits for you to pick it up and carry it all the way to it's specific destination (or better, the cargo station/halt's caption area in which the destination is located.) To my knowledge, there is no time limit, but there is some kind of expiration process running on each cargo platform while waiting.

Yeol, you obviously know much of the game's data structure. Don't you think that the creation of so many new producers and recipients can have a bigger impact on performance? (with mail and waste every town building becomes a factory, sending and receiving products). I just know that path finding is one of the most intense operations in games like this and the performance of TF2 is already critical...

My knowledge of the game's data structure is rather poor. There is so much I would like to do, but I'm not able to, because of my lack of knowledge. But that is not the issue here. ;)

Every town building is already a factory. They all produce and receive several, to even dozens of persons. Adding 1 mail and 1 waste will hit performance a little, but it's participation will stay far below an estimated 10% of the total agents (cargo and people) walking around and finding a path to their destination.

But something had to be done to make the dull vanilla game a little more interesting and varied. Mail and waste are a welcome distraction from the industrial production tree and passenger services. And, the developers are looking with priority how to smooth the game play, and limit the memory consumption. It will all get better. ;)