As a completely speculative answer, I would say:
1) According to our current theories OUR time did start with the bigbang. I say our time, because it is plausible to think of other parallel universes with their own laws and times. Not even so, but no law forbids that other universes have more than one time dimension.
2) You have two possible answers, either the nothingness, no time no space no nothing. Or a pre bigbang universe with its own concept of time. One way to imagine this is to think that you create a very advanced simulation in a computer in which intelligent beings populate the virtual world. Time for them is not the same as time for us. For them their time (that for us can run a different rates if we want, or even stop the simulation and resume it later) is not our time. Their time will start at some point and our time will be equivalent (not the same, as the analogy is loose) to the pre bigbang universe.
3) Time can or cannot have an end, having a beginning does not logically imply it will have an end. However, as with the computer simulation, time can pause or have different rates as measured by an "external" observer outside our universe, but there is no way we could detect that, much less find a proof, as for us time runs always at the same pace.