I want to build an application for a client, I do not have a business registered. Is ok to send an invoice to a client through PayPal and use my own name in the invoice, not any business name?
The USA Small Business Administration website talks about the various possible business types, one being the Sole Proprietor. Here's an extract of the definition of the Sole Proprietor, from here (emphasis mine):
A sole proprietorship is the simplest and most common structure chosen to start a business. It is an unincorporated business owned and run by one individual with no distinction between the business and you, the owner. You are entitled to all profits and are responsible for all your business’s debts, losses and liabilities.
You do not have to take any formal action to form a sole proprietorship. As long as you are the only owner, this status automatically comes from your business activities. In fact, you may already own one without knowing it. If you are a freelance writer, for example, you are a sole proprietor.
You are automatically a Sole Proprietor in the USA. As a result, you are free to invoice clients as necessary.
There are reasons to register as a business. Some people register a company so that they may sell shares to potential investors. Some people register a company because of the limited liability benefits that come along with.
Neither of those are necessary, though. If you're just starting out, there is no problem delaying business registration until you have reason to.
Absolutely. The technical term for doing so is called being a "sole proprietor".
"Sole proprietorship" refers to how your business is "organized" as per the IRS. A sole proprietor may use their own name for conducting business, or may register a business name with their municipality if they want. (My town registers business names for
two four years for $60. With the resulting document, you can get a bank account in your business' name.)