Short answers, pulled from the story.
Compound microscopes first appeared in Europe around 1620, including one demonstrated by Cornelis Drebbel in London about 1621 and one exhibited in Rome in 1624. The actual inventor is unknown, though the optical microscope is the oldest type of microscope.
The true inventor of the compound optical microscope is unknown. Claims have been made for Zacharias Janssen, his father Hans Martens, Hans Lippershey, Cornelis Drebbel, and Galileo Galilei, who built an improved version after seeing Drebbel's instrument in Rome in 1624.
The maximum resolving power of optical microscopes is typically limited to around 200 nanometers because of the diffraction limit of visible light. A new lens using multiple scattering of light has reached resolutions below 100 nanometers.
A simple microscope uses a single lens or group of lenses for magnification through angular magnification alone, like a magnifying glass. A compound microscope uses one set of lenses to enlarge the image produced by another, achieving much higher magnification, and most modern research microscopes are compound.
The total magnification of a compound optical microscope is the product of the eyepiece power and the objective lens power. A 10 times eyepiece with a 100 times objective gives a total magnification of 1,000 times.
Frits Zernike won the Nobel Prize in physics in 1953 for phase contrast illumination. On the 8th of October 2014, Eric Betzig, William Moerner, and Stefan Hell received the Nobel Prize in Chemistry for super-resolved fluorescence microscopy.
Alternatives that do not use visible light include the scanning electron microscope, transmission electron microscope, scanning tunneling microscope, atomic force microscope, and X-ray microscope. They use shorter-wavelength waves to achieve higher resolution but cannot show colour and often require a vacuum.