1) a. What caused the Big Bang to happen? How exactly does matter and energy spring into existence from nothing?
We don't know that the Big Bang did happen.
It is a hypothesis crafted to explain observations that indicted an expanding universe. The only evidence to support the Big Bang is the expanding universe - which creates circular logic.
The Big Bang depends on the expanding universe, which is in question:
{Theoretical and observational challenges to standard cosmology such as the cosmological constant problem and tensions between cosmological model parameters inferred from different observations motivate the development and search of new physics. A less radical approach to venturing beyond the standard model is the simple mathematical reformulation of our theoretical frameworks underlying it.
While leaving physical measurements unaffected, this can offer a reinterpretation and even solutions of these problems. In this spirit, metric transformations are performed here that cast our Universe into different geometries. Of particular interest thereby is the formulation of cosmology in Minkowski space.
Rather than an expansion of space, spatial curvature, and small-scale inhomogeneities and anisotropies, this frame exhibits a variation of mass, length and time scales across spacetime. Alternatively, this may be interpreted as an evolution of fundamental constants. As applications of this reframed cosmological picture, the naturalness of the cosmological constant is reinspected and promising candidates of geometric origin are explored for dark matter, dark energy, inflation and baryogenesis. An immediate observation thereby is the apparent absence of the cosmological constant problem in the Minkowski frame.
The formalism is also applied to identify new observable signatures of conformal inhomogeneities, which have been proposed as simultaneous solution of the observational tensions in the Hubble constant, the amplitude of matter fluctuations, and the gravitational lensing amplitude of cosmic microwave background anisotropies. These are found to enhance redshifts to distant galaxy clusters and introduce a mass bias with cluster masses inferred from gravitational lensing exceeding those inferred kinematically or dynamically.}
Anytime one starts speaking of "settled science" they are promoting faith, not science.
Science is a methodology - not a thing. Science doesn't say anything - it is a tool to organize inquiry in a logical and systematic manner.
1) b. Why were the physical and mathematical properties which dropped out of the Big Bang so finely tuned for the creation of complex matter and molecules?
We don't know that they are. You are shoe horning random data elements to reach a desired conclusion.