Hopefully I can add to some understanding of rivers for the downstream part. My knowledge mostly comes from the Dutch situation.

The most important words for rivers in mountain-like terrain is erosion. Most people know that. Sedimentation is however the word for low-land rivers. Sediment (=soil and tiny rock fragments) is being eroded away by large rivers. However, not all that sediment disappears into the sea.

In low-land, rivers start to slow down, they get a bit lazy. And because this slower flow-rate (less relief) the heaviest sediments start to sink. Particularly when a river overflows it's banks and retreats again. This water on the banks comes to a complete stop. ALL sediment starts to sink down, and the water eventually drains. The banks become 'higher ground'. If this happens enough times, the rivers main stream will get choked off, and the main stream will bend off around the 'tight spot'. The water in the 'old' river will slow down, and drop more sediment. Eventually the old river will dry up completely.

Now, even though all this sediment rained down on the upstream part of the low-lands, once the river drains into the sea, there's still a lot of sediment in it. This tends to fall down close to the river. Basically the same thing happens close to shore as it did inland. The river throws down sediment, and land starts to rise. Once formed, the river will flow around here. Eventually, delta's come into being. Because of this process, as long as the river flows the delta will keep moving into the sea (SLOWLY), and branches on the main river in the OLD part of the delta will appear as major rivers branching off.

Everything in geology is dynamic, it only moves slower than our lives. For things like geology you need to put away your human time-perception, and switch to one where 500 years is 'a short time'.

Now, all this would be fairly simple if we didn't exist. Rivers follow their own rules, but humans can use those rules. By carving out a channel, a river is easily persuaded to flow into another river. This happens mostly around Delta's I guess, but it makes the situation a bit more complicated. Man, likes having a boat. But man also likes his home. He would also like to have his boat close by his home. Now this is fine, if only the damned river didn't move off... So man will ensure his access to the river by either guiding the river, or by digging channels. This will make us cartographers completely confused about what part of a river is man-made (or man influenced) and what part not. Since our ancestors didn't really care about perfectly straight waterways. Some waterways follow already existing low points in the landscape. Making them look quite natural.

To name a few rules:
- GENERALLY rivers only join. They don't separate (delta's are the exception).
- Rivers flow to somewhere, this may be a lake (which may, or may not drain into another (or the same) river, but this is usually the sea.
- Rivers flow faster upstream than downstream.
- In delta's weird stuff can happen, like rivers branching off into two smaller rivers, only to flow into another river, or another earlier branch. As a rule of thumb, delta's are quite big if they extend about 100 km (about 60 miles) inland. Generally the bigger the river, the bigger the delta.
- Meandering (squigglyness) takes place DOWNSTREAM rather than upstream. The slower a river flows, the more it meanders. Upstream where it flows faster, it usually is much more 'straight'. To get a good feel, go to Google earth/maps, and follow some different rivers in their flow. You'll get a good feel eventually. It isn't really straight what I mean, but you'll see the difference.
- Don't forget, unless you build a planet without intelligent life, a river will be influenced by intelligent species. As a rule of thumb, if a civilization has bigger ships, it will more actively influence it's rivers. And with 'bigger ships' I mean bigger than a canoe. Draft-ships (ships being pulled by horses) for example. In 'modern' societies (industrial revolution and later) this process only expands. There however will always be people who leave the river as is. Once again, I'm basing this mainly off the Rhine.
- Two or more rivers can share a single delta.
- In more local maps depicting a river delta, rivers CAN and WILL branch off and join together again.
- River islands can be natural, or man-made (channelized river).
- Rivers don't flow from one ocean to another.

Basically, the mentioned map is a fine example of how it's done wrong. I doubt it was made this way for this purpose, but showing how it's not supposed to be done is a good way of teaching people