The success of advanced quantum communication relies crucially on non-classical light sources emitting single indistinguishable photons at high flux rates and purity. We report on deterministically fabricated microlenses with single quantum dots inside which fulfil these requirements in a flexible and robust quantum device approach. In our concept we combine cathodoluminescence spectroscopy with advanced in situ three-dimensional electron-beam lithography at cryogenic temperatures to pattern monolithic microlenses precisely aligned to pre-selected single quantum dots above a distributed Bragg reflector. We demonstrate that the resulting deterministic quantum-dot microlenses enhance the photon-extraction efficiency to (23±3)%. Furthermore we prove that such microlenses assure close to pure emission of triggered single photons with a high degree of photon indistinguishability up to (80±7)% at saturation. As a unique feature, both single-photon purity and photon indistinguishability are preserved at high excitation power and pulsed excitation, even above saturation of the quantum emitter.