For designers, the client is usually the company they work for, rather than an outside client. For the purpose of answering your question, let’s assume the designer is a freelancer working directly with a client. Even in that scenario, every situation is different, so I’ll answer you question in very general terms.
Maybe in crowdsourcing situations, but that’s one of the problems with crowdsourcing: clients don’t have a clue how to write a brief that covers everything the designer needs to know.
A better approach is for the designer to talk with the client about the project. This gives both a chance to clarify the project’s details by asking questions, raising concerns, and discussing issues that are unique to each project.
In a traditional advertising agency, the process always works this way. The account rep (and sometimes the creative or art director) discusses the project with the client. In this situation, the account rep, in collaboration with the creative or art director, writes the creative brief for the designer.
As I mentioned, though, every situation is different.
In a freelance situation where the designer is working directly with the client, that’s possible, but I would never do that except in very rare instances when I’ve developed a tight collaborative relationship with a very savvy client.
Showing a sketch or rough idea to clients is almost always interpreted as an invitation to collaborate and interfere in counterproductive ways. Clients aren’t designers, and they can’t typically see past the roughs, so they begin giving unhelpful feedback based on misinterpretations of what they’re seeing.
This is even worse if the rough ideas are sent to the client by either mail or email. In that situation, the same thing inevitably happens, and it’s never good. In that situation, the typical response from them is something along the lines of, "I’ve been thinking about your ideas all day. I showed them to my wife (or husband, or best friend, secretary, or guy I met on the street), and we like the symbol you drew, but instead of it being so abstract, could you draw a fun cartoon monkey hanging from a tree while eating a banana?
In other words, the foolish decision to mail a rough idea to a client has turned into the disaster of the client asking you to draw a stupid monkey.
In my opinion, the first glimpses of your thoughts should be in the form of very tight mockups (comps) that are so polished, well-done, and immaculate that they leave no room for misinterpretation or client modifications. However, clients always want to feel like they helped, so we would always have a set of questions whose answers made little difference to the design, but satisfied their need to feel included in the creative process.
Much depends on the budget. On some projects, expensive working prototypes are warranted. On others, a tightly rendered drawing might suffice. No matter the budget, the objective is to communicate the solution to the clients in ways that favorably impress and convince them, and that gives them the confidence to give you the go-ahead to proceed.
After the job is done and the client is happy, a small sales pitch for whatever else the client might have in mind down the road might follow.
Payment generally depends on the contract. If you’re working with a government agency or a large corporation, you’ll likely agree to their terms, which usually involve payment after the job is completed or in monthly installments for long-term projects. Sometimes, with reputable small clients, a series of emails outlining the project and payment terms are all that’s needed, as emails can act as informal contracts. For new clients, especially if there’s a chance they might be unreliable or dishonest, it’s best to break the project into phases and require payment in advance to start each phase. For instance, pay upfront for the ideation stage and mockups, then pay in advance to begin implementation. For these newer, higher-risk clients, full payment should be collected before the final delivery.
As I said, every situation is different, and everything I’ve mentioned has exceptions, but the objective is always to impress clients and get paid.
I generally agree with what @Smurf2 wrote, but unlike him, I’ve never shared rough ideas with clients for the reasons I mentioned — either in freelancing or in my agency roles. Again, though, this is an example of what works well for one person or company but not for another. There is no universal one-size-fits-all best way of doing things.