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What might replace the cosmic singularity in a correct physical picture of the big bang? Will it be the emergence of time? Or a bounce? Or a conformal match?

If we accept the standard rules of quantum mechanics, embedded into the framework of string theory, then the singularity is certainly resolved. We don’t have a non-perturbative definition of string theory, but already perturbaive arguments seem to indicate that the infinite tower of states present in string theory is enough to resolve the singularity.

There are other proposal besides string theory, such as eternal inflation.

In fact, eternal inflation does not resolve the problem, since the `eternal’ refers to future, but if you extrapolate backwards in time, you obtain a singularity at a finite time.

This question is also closely related to the first one since it certainly concerns quantum gravity and perhaps foundational aspects of quantum theory itself.

Loop quantum cosmology. It is capable of coming up with consistent cosmological backgrounds including post-Einsteinian corrections.
Perhaps the information from gravitational waves measurements is going to shed light, pose constraints.
We have no idea.  How would you know experimentally?

Theoretically, there are no satisfactory answers.

Maybe if we have a theory quantum gravity.

Regardless of what the replacement would be, it is important to have a way testing these ideas.
     - For example, if inflation is true and smooths out early spacetime structure it could be difficult to distinguish between competing models

Some of these ideas are a bit naïve in that they are based on hypothesis which will likely be shattered by whatever theory supplants our current ones.

With regards to the first point, CMB and gravitational wave raw data could give us some access to very early times.

Could we test some of these ideas by creating a black hole/strong gravitational field.

Given that the very early universe was in a state of constant quantum fluctuations how to achieve a classical world, i.e. did the universe measure itself?

In an n-body problem with simultaneous ("total") collision, you can't extrapolate the trajectory past the total collision.  Does this example bear some relationship to discussions of conformal matching and Penrose's cyclic universe models?
Are clocks an operatioanally valid concept in GR?  Here's one example of an operationally-defined clock. Take a drop of fluid in another miscible fluid, and characterize how these mix in the laboratory. Share this characterization with other observers. Now let this clock wander off (near a black hole, for instance) and let yourself or other observers re-collect the clock and observe its degree of mixing to measure the proper time experienced by the clock.  But is that not a fundamental clock?  (Because it makes use of classical physics and observers?)  Meanwhile, could quantum fluctuation and correlations be used as a clock?  What happens when any of these clocks fo into a black hole?  Do clocks entering black holes burn in the firewall? Is there simply no possible operational clocks in some situations, such as in the early universe, as suggested by Missner, Thorne, and Wheeler? 
Any big-bounce-like description of the universe, when abstracted, looks like Poincare recurrence. Imagine balls moving in a box.  They may end up at a point, and then they end up at the point again. Could this be something like what our universes is doing if its cyclic?  With such questions, we must keep in mind whether the phase space of the universe has bounded measure (or else Poincare recurrence does not occur).

Do we know there was a big bang? What about eternal inflation?

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