Because this is a simple statement, and Italian often uses the same basic order as English:
So the structure is:
subject + verb + place
Italian can be more flexible than English with word order, but this is the most neutral and natural order here.
Yes, you could. Sulla trave and sopra la trave are both possible, but they are not always felt exactly the same.
In this context, if the object is physically resting on the beam, sulla trave is a very natural choice.
Yes, they are related in form, but they are not interchangeable.
So in a sentence about a tool resting on a beam, livella is the right word.