Short answers, pulled from the story.
Mach's principle is the hypothesis that the large-scale distribution of matter in the universe determines what counts as rotating and what counts as stationary at any local point. A very general statement of it is that local physical laws are determined by the large-scale structure of the universe.
Albert Einstein coined the phrase Mach's principle. Ernst Mach himself never stated the principle explicitly; Einstein named it after finding inspiration in Mach's 1883 book The Science of Mechanics, and Einstein was the first to use the term in the literature.
Ernst Mach published the key ideas in The Science of Mechanics, which appeared in German in 1883 and in English translation in 1893. Before Mach, similar ideas appeared in George Berkeley's De Motu, and after Mach, Benedict Friedlaender and his brother Immanuel published related arguments in their 1896 book Absolute or Relative Motion?
The Lense-Thirring effect is the precession of an inertial frame inside a spinning spherical shell of mass relative to a fixed background. Einstein found it before completing general relativity and interpreted it as evidence for Mach's principle, writing to Mach that a Coriolis force arises inside such a rotating shell and that the plane of a Foucault pendulum inside would be dragged around, though with a practically unmeasurably small angular velocity.
Most physicists say no. At an expert poll held in Tubingen in 1993-22 respondents said general relativity is not perfectly Machian and only 3 said it is. With appropriate boundary conditions involving spatial closure, 14 of 21 respondents agreed general relativity is very Machian, suggesting the answer depends heavily on which formulation of the principle is used.
At least 21 distinct formulations are possible in the context of general relativity. Hermann Bondi and Joseph Samuel listed eleven statements they labelled Mach0 through Mach10, taking inspiration from the Mach number, though they noted their list is not exhaustive.