Are quantum events such as decay, emission, absorption and tunneling truly instantaneous or is there some small time period? Are quantum events such as decay, emission, absorption and tunneling truly instantaneous or is there some small time period?
Assuming there is an underlying cause for a quantum event it seems to me that we should expect some time interval between the cause and the effect even if the time intervals are too small to be experimentally measured.
Consider a magnet moving very gradually towards an iron filing. The iron filing is observed to start moving towards the magnet until it is drawn rapidly together with the magnet. The elapsed time of the event is the time between the iron filing beginning to move and being stationary with the magnet.
 A: The average delay before a decay or emission event is known as its "lifetime" (or, with a factor of $\ln 2$, its "half-life"). Quantum states with the same quantum numbers are indistinguishable. There is no difference between a system which happens to be on the cusp of a decay versus a system which happens to end up not decaying for many half-lives.
A: It depends on one's perspective.
We never start in an eigenstate
If we are solving a Schrödinger equation for the eigenmodes, the solutions will already contain the particle on the two sides of the barrier or a photon simultaneously emitted and non-existent (with the atom in the excited state). Note that I am talking here not about superpositions, but about the eigenstates of the full Hamiltonian.
In practice we never start with an eigenstate, but with a superposition of them: as an electron localized on one side of a barrier or an atom in the excited state. We then observe the system evolve, and never return back to its original state _ either because the interaction with the environment causes decoherence at some point or because the time between the collapse and the revival is too long for actually observing the revival. We then talk about a tunneling event or an emission/absorption event, and we can attribute to them a characteristic time.
Mathematically we often treat such events in perturbation theory, e.g., considering an isolated excited atom and isolated EM field, and then turning on the perturbation and observing how the atom excitation "leaks" into the field. Gamow's calculation of the radioactive decay is a classic here (which has been adopted to other situations, e.g., to tunneling in quantum dots by Gurvitz, and extensively used.)
Beyond Quantum
Note that there is nothings pecially quantum about this:

*

*same effects can be observed we discussing classical electromagnetic fields (i.e., the stationary and dynamic solutions of Maxwell equations).

*in thermodynamics/statistical physics we encounter extreme situations, where the system does not have time to explore all the states of the same energy and, contrary to our expectation, finds itself in one of these states for seemingly infinite time. This is what we refer to as spontaneous symmetry breaking. A required reading here is the Anderson's More is different.

