So if I want to trade on the exchange, I first need to buy BKC which I then can *consume* in my trades, correct?
That's right. All network messages require a fee to give robust protection against denial of service attacks. We expect to make acquiring your first BlockCredits painless by integrating support for Shapeshift or similar exchange service seamlessly into our user interface. So, the user will request a BlockCredit and will be asked to send a tiny amount of Bitcoin (or other cryptoasset) to a particular address. When they do, they immediately receive the BlockCredit they need to use the network.
While Shapeshift has advantages for some use cases such as the one just mentioned, B&C Exchange is likely to offer better pricing for the most popular trading pairs, permits limit orders (placing an order an that won't fill now but may fill later if the market price changes) and is expected to permit much larger trades (after a beta period).
Now the other side: Someone applies for a custodianship on the exchange. BKS shareholder approve the proposal and grant the custodian 1000 BKC. The custodian now sells those BKC (in a centralized way?) externally to the exchange and passes the revenues back to the shareholders in form of BTC. Correct?
That's right. It is expected that BlockCredits will be available for sale on B&C Exchange, traditional centralized cryptoasset exchanges and instant exchange services such as Shapeshift.
Is there an incentive to act as a signer? Will they get compensated (through a part of the tx fee in BKC) for their availability and reputation?
Absolutely! In addition to the block reward given to minters, there is a second block reward that will be given to reputed signers in proportion to their reputation. The size of the reputed signer block reward is configurable through shareholder voting, so it will be whatever is required to get quality signers and no more. While shareholders may be tempted to upvote their own reputation to receive this reward, other shareholders will thwart this by downvoting.