I've never heard of any agency taking a cut from advertised rates, their margin is already baked into a quote. That said, you do have to run some math on the quotes to get a real feel for what they mean. PanTravelers has a free calculator to help you compare quotes. Some of the issues are that much of the quote is free from income taxes, legal if you maintain a tax home. Some agencies will quote "take-home" pay from gross pay which does some of the math for you - similar to what PanTravelers calculator does. Others quote a taxable hourly pay, and lump the non-taxible housing and per diem stipends into a weekly number.
If you are not eligible for the tax free money, then the quotes are without meaning to your personal situation.
You are considering working for a single agency. This can be a highly effective strategy and reduces paperwork and hassle but I don't recommend it long term. I did this myself for over 4 years for the then largest agency, Cross Country. While I was working for them, it seemed good, I was treated well and had escalating pay (sort of a normal travel progression because as you demonstrate your value as a successful traveler, you get access to better assignments). When that relationship came to a crashing end, it became very clear I could do better in a number of ways.
For one thing, the benefits of managing my own career more closely became apparent when I became a "free agent". Pay escalated even faster when I could freely pick assignments from different hospitals and utilize smaller agencies with less overhead (meaning they could run with a lower margin).
I also discovered the benefits of maintaining my own professional portfolio, especially written references/evaluations. Agencies do collect these from each assignment, but typically consider them proprietary and do not share with their travelers. Proprietary because if a traveler has them, they can easily move from one agency to another with little friction and pick the best paying agency for particular assignments.
That said, working say for the first year for a single agency lowers stress while you learn how everything operates. But my normal recommendation is to talk to lots of agencies and pick maybe three of the ones with recruiters you communicate well with. Then you have a plan B if your first choice goes bad. In fact, my communication with my first recruiter was so bad, I almost bailed before I started. But for several reasons, they were my only choice (in part because I pre-interviewed at a hospital on a holiday I really wanted to travel to and found that they primarily used that agency). So I called them and was able to talk to a recruiter manager and got a new recruiter. The importance of a well fitting recruiter cannot be understated, far more important than the agency brand.
"Settling in" to a new assignment can be tough but not too dissimilar from any other job. Sure, there can be cliques and issues with new policies and procedures, expectations, and new medical software to learn. It can be a lot for a new traveler. Thus it is really important to get a traveler friendly assignment well within your clinical abilities the first time or four - be suspicious of high paying assignments in the beginning - high pay should not be your primary initial goal for well fitting assignments.
Maintaining a high level of professionalism is super important to successful travel. You have to overcome staff distrust of the skills and teamwork of a new colleague, especially that of a traveler. You do not have the "get out of jail free card" that a staff member will have from even what you might consider innocent remarks. Do not criticize their practices (but don't do anything you consider unsafe either), or compare it to them with other places you have worked, or show them a better way to do things. Instead, work hard, be cheerful, and help others when you can. Staff expects really good performance from travelers who make "twice" as much as they do - we don't other than some crisis assignments - but that is what they believe.
Good luck! If you are a good fit for the travel lifestyle, it is a great life!