They help balance the benefits of corporations, make corporate competition a bit more interesting and challenging, help include every resource in the system, and act as a late game reduction to economic growth (that targets successful civs over struggling ones).
About Syndicates: I guess I should avoid to create those syndicates that deal with those resources I own. So I founded The Triads, since I have few of those resources this syndicate deals with. And then I should expect this syndicate expands over AI cities while avoiding my cities... and my benefit is exclusively to bother my opponents. Isn't it?
Pretty much. They're not necessarily something you want to found if you're doing well, unless your opponents will be affected worse than you. They're probably most useful if you're behind and want to destabilise a stronger opponent a bit.
Agents and Executives are deliberately expensive as I wanted Syndicates and Corporations to act outside of direct player control. You can direct them a bit, but it's costly. Can be worthwhile if you've got a lot of shares and the share price is high though.
Second: I've got a quest where I'm tasked to attack and takeover from opponents Silk, Amber and Jade (all of them resources related to the Syndicate The Triads). I feel it not coherent, since I should prefer not to own those resources in order to don't get The Triads expanded over my cities.
It seems the Common Market agreement (diplomacy) does not work properly.
If you open the attached file and press "pass turn" button, Israel will break the Common Market agreement we were involved in. Then suddenly I will lose ALL of the resources I had, even all of those that were on my territory.