We have the music of Palestrina, Gallus, Lassus, Cavalieri, Lechner and others. Most chords are in root position. But some are not. In which period the first inversion of triads came up? Was it right after the organum with the first third chords? what rules were used in 4-part practice? who was the first that established some rules about this question?
That's a great question that's often insufficiently covered in music history classes. Franco of Cologne's textbook Ars cantus mensurabilis in the 1200's declared the major and minor 6ths (at least in 2-part counterpoint) to be dissonances, an even more restrictive category than the 3rds, which were considered "imperfect consonances." This indeed matches French practice at his time: while 3rds and root-position triads were rather freely used (although cadences had to end on an octave or fifth), the 6th and the 6/3 and 6/4 chords are usually kept on weak beats and used as passing tones, etc.
The English were rather freer than this, and often used parallel 6ths in 2-part writing (the Oxford Book of Carols has some early examples) as well as faburden/fauxbourdon, a 3-part style with parallel 6/3 chords ending on an 8/5. In the next two centuries these English practices began making their way through Europe. By Dufay's time (mid-1400s), the 6th was universally heard and taught as an imperfect consonance, and the Lydian faburden cadence Em/G - F5 (to borrow rock notation!) was commonplace. When this cadence is used in a 4-part piece (e.g. Machaut's Messe de Notre Dame), one upper voice is doubled, in spite of the resulting parallel octaves and fifths! This practice declined in the 1400s.
Interestingly, during this time, while the 6th and 6/3 chord was getting more freely used, the 4th above the bass, appearing notably in the 6/4 chord, was increasingly heard as a dissonant tone in need of resolution to the 3rd. So in Palestrina and his contemporaries (excepting the dilettantish Henry VIII!) when a 6/4 chord occurs, the 4th will always be prepared as a suspension, or the chord will arise from passing or neighboring tones, etc. Or, once or twice, he'll lean on the dissonance as a special effect (usually with words like "O cruel death" or "We have acted unjustly" - see Jeppesen's The Style of Palestrina and the Dissonance for more on particular rules). The doubling of the 6/4 chord is limited by the treatment of the dissonance: if the 4th is a suspension, it cannot be doubled without creating parallel octaves, because it must resolve down. Doubling in 6/3 chords in the Renaissance is an incidental by-product of melodic motion; the doubling is not governed by rules, except that chromatic leading tones (e.g. C# in D Dorian) are at this time never doubled.
This is the story of these chords in educated, mostly vocal music. However, the 16th-century Spanish guitar had a reentrant tuning, like the modern ukulele; the extant texts make clear that street musicians played chords on it without regard to which note was on the bottom, which must have driven learned listeners nuts!