Well, these are indeed very valid questions, and founding of an official club may not be as easy as it may sound. The exact procedure differs in each country, so I cannot tell you for the USA, but you will likely need to register as a non-profit association. I think that any commercial lawyer will be able to help you out with the necessary paperwork and it should not be too difficult or expensive.
After a brief googling I found the following document about founding a non-profit organization in Michigan. I guess it won't be much different in other US states either.
https://www.mnaonline.org/pdf/GettingStartedSummary.pdf
Here in France, the club must have management of three persons - the president, the secretary, and the treasurer. I believe it may be similar in the USA too.
In the USA, you probably do not need to register with your national freediving federation, but I definitely recommend it. They may help you with some advices or tips, and with the organization. They will also list you on their website and make it so easier for others to find and join you. Being covered by the federation will also help when negotiating with city or state offices - for example when trying to get some hours/lanes in public pools for training (padding pool is also a good idea and usually easy to reserve some evening hours for wet statics training).
Having a room for meetings, theory seminars, and/or for stocking some material is not bad, but not absolutely necessary.
You are certainly not obliged to organize any competitions, or participate on them, but usually at least some members will want to do it later or sooner anyway.
As for liabilities - again that differs in each land, and consulting it with a lawyer (and with the federation) may be the best approach, but you need to count with at least some degree of legal responsibility at the club management, and at trainers / instructors. Here in France the instructors must be certified by FFESSM (our federation), and we all are covered by insurance which has a contract with the federation.
You will also likely need to file tax declaration, so the treasurer should have some experience with accounting (properly filing member fees, and club expenses).