1 CDSs are usually traded in notionals no less than 5 million, usually much larger. Most of the volume comes from
CVA desks of institutions that try to flatten their counterparty exposure
CDS market-making desks that generally don't take views, but try to make money from bid-offer spreads
hedge funds that can take views on CDS itself or relative avlue of CDS and other instruments, but generally get paid for providing liquidity.
they all have ISDA agreement with each other. These are over the counter (OTC) swaps. There are efforts to bring all CDS to swap execution facilities (SEF) that will eventually succeed.
There have been many attempts to create retail products (smaller notionals; notes that don't need ISDA agreements or exchang-traded) whose payout would be linked to CDS. I am not aware of any very successful ones so far.
As to who ends up with non-flat CDS exposure at the end of the day, most commonly CVA desks try to have the CDS exposure that would offset the unwanted credit exposure that their firm has from other relations with the counterparty.
very large reinsturance firms (based in Nebraska, Switzerland, or caribbean) are sometimes very large sellers of CDS protection. (Don't worry, it's no riskier than being long a lot of bonds... almost :)
A couple of other made-up examples
suppose a large corporation (not financial) is doing a multi-year project in a made-up Kingdom of Povonia. They're worried that political turmoil in Povonia will force them to cancel the project. They mitigate this risk by buying CDS protection. Note that they don't to wait for a credit event that would cause CDS to pay out. Under the scenario, they cancel the project and unwind the CDS, which would be much more valuable because of the turmoil and help offset their losses.
suppose a pension fund or mutual fond is long a lot of risky bonds that are not very liquid. They want to reduce the exposure, but they're afraid that if they sell the bonds, they won't get a good price, and if they want the bonds in the future, they will be expensive (imagine 5% bid-ask spread). They mitigate the exposure by buying CDS protection. When they want the exposure back, they unwind the CDS. This is cheaper than trading the bonds if the CDS is more liquid than the bonds.
2 in a standard CDS, the payment for the protection is fixed from the inception to the end. The protection buyer pays X. If later the same maturity protection is trading in the market at Y > X, it's good for the buyer and bad for the seller, but the cash flows don't change. Actually, in the standard CDS, the protection buyer pays (or sometimes receives) a substantial upfront fee, and then pays a standard running spread, typically 100 bps.
There used to be a version of CDS called Constant Maturity CDS (CMCDS). It was pretty unusual before 2008 and I'm not aware of any printed after 2008. Suppose the protection is trading at X. For the first period, the protection buyer pays p * X, where p is fixed "participation" (say, 80%). Also, no material upfront fee. Suppose in the next period we observe (for example on IHS Markit) that the cost of 5-year protection has moved from X to Y. The buyer pays p * Y. However there is always a cap, like 700 bps. (Note that you need to consider the implied volatility of the credit in order to price this.)
3 are rough approximation. In particular, in PL in case of default includes not just notonal - recovery, but also the pv of the CDS contract (which can be substantual if the protection was bought when it was much cheaper). This CDS spread is an approximation of what the protection would cost without an upfront fee. In reality, there's an upfront fee and a stanard running spread of 100 or 500 bps.