Whether the is required in your examples depends on the context:
It was made by machine
is a way of saying that it was made mechanically and not by hand.
It was made by a machine
means that it was made by a machine that has not been identified.
It was made by the machine
means that it was made by a machine that you have already identified.
You cannot say: It was designed/built by architect without an article.
Either: It was designed by an architect (who has not been identified)
Or: It was designed by the architect (whom you have already referred to)
Thus you could write either: It was designed by John Smith, (who is) an architect, or: It was designed by John Smith, the architect (which assumes that John Smith is well known as an architect.
The same rule applies to band. Music is not performed by band, but either by a band (unidentified) or the band (already identified).
I think there is a complication here because both "tripped on" and "tripped over" can have multiple meanings:
"tripped on" can mean either that the mentioned thing was the thing that caused one to trip, or it can just mean that one was "on" (on top of, or at the same location as) the thing when one tripped, so "tripped on a cable" probably means that the cable caused you to trip, but "tripped on the stairs" doesn't necessarily mean that the stairs caused the trip, you might just be saying that's where you were when you tripped.
"tripped over" can mean either that (again) the mentioned thing caused one to trip, or it could just mean that when you tripped you went over something in the process of falling, etc. So, "tripped over a cable" probably means that the cable caused the trip (the same as "on"), but "tripped over the wall" probably doesn't mean that the wall caused the trip, but rather that something else caused you to trip and in the process of tripping, you ended up going "over the wall".
So both "trip on" and "trip over" can mean that the thing caused one to trip, or they might not. Which one is meant is often a bit ambiguous, but usually not that hard to figure out from context and what the thing in question is.
Best Answer
"One out of many" implies "One randomly chosen out of many similar ones."
"Choose one from many" has a similar meaning.
"Out of many, one" means that "Many were combined, to make one." (It is a translation of the Latin phrase e pluribus unum, which is the motto of the United States of America. Under the doctrine of dual sovereignty, the many continued to exist in parallel with the new federal government.)
"One from many" is ambiguous. It could mean "One chosen from many", or "Out of many, one."
The original poster's fourth through seventh questions can be answered affirmatively.