As mentioned in the comments, I think the U-turns of the river in your example board are invalid; see (this version of) the rules. If, as you say "We must also exclude possibilities that become unplayable when the river curves in onto itself.", the Java program at the bottom will calculate all possible boards, with the caveat that it distinguishes between ...
The list of expansions that add pieces for each player are as follows:
7 meeples, plus one for scoring.
Inns & Cathedrals
Adds the Big Meeple.
Traders & Builders
Adds the Builder and the Pig.
Abbey & Mayor
Adds the Mayor, the Barn, and the Wagon.
Bridges, Castles, and Bazaars
Adds 2-3 bridges and 2-3 castles, depending on the ...
I accept the answer, but doing a little bit research, I found in the Complete Annotated Rules the following:
The original starting tile can either be mixed into the stack of river tiles, or placed as soon as the river has been completed, and is treated like a normal tile.
And an additional house rule:
Mix the original starting tile (without a river) ...
(adding to Kristo's answer because this is a bit too long for a comment)
We regularly play "SuperUltraMega Carcassone" because we have most of the expansions and two copies of the basic game and we are, frankly, silly. It can take a couple of hours to play that way, but it's kind of fun and means we only have to clean up once. Plus there's the challenge of ...
Sometimes these rules can be confusing, but the iOS edition of Carcassonne has answered a lot of rule grey areas for me, including the River II city-bridge one. The Carcassonne app makes it clear that this city-bridge over the river is, in fact, high enough to keep the fields connected.
While I still find this visually confusing sometimes, the app rules ...