In quantum physics, a particle can be in a superposition of two states
until it is measured. In other words, the aforementioned particle
doesn't have a definite state until it is "looked at" (measured).
A superposition state is a state. So a particle in such a state does have a state before being measured.
Since certain properties (i.e., an electron's position) of a particle
aren't well defined until they are measured, does this mean that
quantum objects don't possess these properties, unless they are looked
No. It means that the particle doesn't have a single position. Rather there are multiple instances of it at different positions. Those instances can interact with one another in interference experiments, which is why you can't understand what it is doing by considering it as having only a single position. See
Do quantum objects even exist without an observer? Since macroscopic
things are made of quantum particles, does the moon stop existing when
no one is looking at it, since its particles don't have a definite
position, momentum, etc.?
For a large non-isolated system like the moon, interactions with other systems prevent it from undergoing interference on a macroscopic scale. It does exist in multiple versions but you can't see the other versions, see
Do the answers to these questions depend on the interpretation of
quantum physics being used?
Different "interpretations" of quantum mechanics make different claims about what is happening in reality. So they ought to be treated as different physical theories, not interpretations. Quantum mechanics without the collapse postulate implies the Everett interpretation:
Other interpretations may have different implications about how the world works but they are usually ambiguous about what exists and some are apparently inconsistent or otherwise unviable.
I guess my main question is this: Is there an external, objective
reality that exists even when it is not being measured?