does the end client usually know/care how much the consulting company
If the client wants to know, they will know.
Occasionally a client company will care. Most often they do not.
Clients tend to focus on their cost. You need only focus on your pay. If you get the pay you want, it shouldn't matter at all how much the client ends up paying. Whenever I've been a consultant through a consulting company, I just told the consulting company I worked with "I want $x". It was up to them to figure out how to get it for me. If they wanted to get a huge market, good for them. If they needed to lower their markup, too bad for them.
If you want total control over the client's cost as well as your pay, then become an independent and avoid the consulting companies completely.
If the end client decides at some point to lower how much they are
willing to keep paying to the consulting company, do they usually tell
the consulting company to lower accordingly my actual pay, or is
something they leave to the consulting company to decide about?
If the client wants to lower their cost, they certainly won't target your pay. They just decide what they want, then it's up to the consulting company to decide how to deliver that.
Some consulting companies will lower their markup. Others will just seek out cheaper contractors.