What defines the observable universe and why is its radius about 45 billion light years?
The observable universe is the sphere from which light has reached us since the Big Bang; because space has expanded since that light was emitted, the most distant things we can see are now about 45 billion light years away.
How can the universe be finite but have no edge?
If 3D space is curved back on itself (like the 2D surface of an orange mapped to higher dimensions), it can have finite volume without a boundary—travel straight and you'd eventually return to your starting point.
What observational signatures would a 'hyperdonut' (toroidal) topology produce?
A toroidal topology could produce repeated images of the same object in different sky locations and at different apparent times, causing a cosmic 'hall of mirrors' effect.
Does an infinite universe guarantee duplicates of people and planets?
In principle yes: with a finite number of particle configurations, an infinite space implies repetition; however, exact duplicates would be astronomically far away and effectively unobservable.
Can current observations tell us whether the universe is finite or infinite?
Not definitively—searches for topological signatures (like matched circles or repeated patterns) have found none, so if the universe is finite it must be much larger than our observable patch.