There's a reason there are so many inversions in this particular piece: notice that the bass is in parallel sixthsthirds with the melody the entire time!
As for an inversion appearing before the root-position version of a given chord, that's not unheard of or rare by any means; it's actually relatively common.
And you're correct that you usually don't omit the third. But in a piece with as sparse a texture as this, occasionally composers will bend the rules a bit, and this is one of those instances; we still understand this chord at the end of the second measure to be a V43 even if there isn't an F♯ in it.
(PS: This is great repertoire that I didn't know; thanks for introducing it to us!)