Translation:The farmer is not in the theater, but in the field.
My understanding is that yes (so someone correct me should I be mistaken), ⟨th⟩ would normally be pronounced /tʰ/, as it was a transliteration of ⟨θ⟩ from Ancient Greek (as you said, before /θ/ emerged).
I remember seeing something saying that it was likely that ⟨ch⟩, ⟨ph⟩, and ⟨th⟩, though all aspirated, would have likely only been aspirated by educated individuals since they represented 'foreign' sounds. But I can find were I saw that.
I do not think there would be a difference in Latin. Maybe in more specific contexts.
I have definitely heard both, but in different contexts (this of course likely depends on where you are and what dialect is being spoken).
I tend to hear 'on the field' when it come to sports/games or combat (but here usually hear 'battlefield').
In most other contexts I can think of, like farming or just general location, I usually here 'in the field'.
But of course I do not speak for the entire English speaking population.