Theranos Inc. misled company directors about its laboratory-testing practices, used a shell company to “secretly” buy commercial-lab equipment, and improperly created rosy financial projections for investors, according to allegations in newly unsealed court filings in a suit by one of its investors.

The Silicon Valley company—which once promised to revolutionize the blood-testing industry using tiny samples from finger pricks—also ran “fake ‘demonstrations tests’ for prospective investors and business partners” using commercial...