Pretty much about right. In this example, MS is a prime broker...

Pretty much about right. In this example, MS is a prime broker with a number of HF clients, the first HF may have decided to short something else and covers, returning stock to MS, which it can lend another of its customers or return to JPM. JPM is the custodian for Blackrock in this example and it wouldn't return stock to Blackrock (Blackrock is using JPM as custodian rather than doing it itself). They would know that their stock is available for lending and would be able to enquire about whether stock is out on loan or not at any particular moment.

Yes it is finite in the sense that our happy HF can only short stock they have borrowed once. If they want to short more, they have to borrow more by asking MS for more. If MS doesn't have any, they can ask JPM or any other stock lending agent. Some HFs use more than one prime broker, but the process is the same: borrow from the PB, who either has some or has to source it from someone else. The reason it is finite is because when you buy a stock, you expect them to give you the stock! I can't deliver you a stock that I sell you unless I have one to give you.

Eventually, all stock in all lending programs can be used up so that no further shorting can take place unless some gets covered / returned to a lender. The lenders know this, so high demand stocks get a higher borrow fee than other stocks (most stocks have plenty of slack available). GXY is a high demand stock that you can expect to pay 8-10% annualised, accruing daily on the value of the stock outstanding.