Time is a property of vibrations (space-time). To explain waves the ether was introduced. The ether does not have references and physical quantities need references to assign values. It's known that waves are energy and energy is conserved. If energy is conserved the ether must be perpetually filled with energy of waves that cannot get less or more. Oncoming waves become vibrations and vibrations tend to arrange into vibrational pattern. The vibrational pattern are quantizing the ether into space units and time periods which generate the absolute space-time units. So far, absolute space-time had not been measured because observation is based on matter. Matter is just modulations on absolute space-time which means that they are modifications of the vibration pattern. Wave equations indicate that the modulations change with velocity. Trying to measure how much space and time changes when matter changes velocity fails because the references change accordingly. But if we use references from some matter and measure the values from matter with a different velocity it shows up that space and time changes as expected.

Time would have to exist in the voids of space. Gravity exists crossing voids.

It would be very strange if time did not exist there. And even interstellar space isn't really a void; there just isn't much stuff there.

Depends on what you mean by "time". In the physics sense, yes, absolutely.

No.

With a volume that lacks much matter, (I'm not sure that there are any perfect voids in this universe, strictly speaking) on a statistical level the interactions between that matter would be less and therefore the behavior of that matter would be more reversible than in areas of more concentrated matter. In this sense, related to the entropic arrow of time, time would matter less in a statistical physics sense than on a planet or star (though it would still factor into considerations).

General relativity is just fine with empty space. In a mostly void volume of space, space will be much less curved by the matter and as such, time will progress more slowly as well.

In quantum field theory, from what I understand, time still applies even in a vacuum.

So the short answer is yes. But it would be at least a little different than the way time is treated for a person here on the surface of the Earth.