Q:

What is the best explanation for the fine tuning of the Universe? 

In my view, this is one of the most profound unsolved mysteries of Science.  How to explain the fine tuning of the Universe without recourse to saying that a God/Designer did it. 
Examples of fine tuning:
We exist because of a crucial one part in a billion difference that favors matter over antimatter. 
If the ratio of the nuclear strong force to the electromagnetic force had differed by 1 part in 10 to the 16th power, no stars would have formed. 
If the strong force constant were 5% weaker, there would be no stable stars
If the strong force constant were 2% stronger, there would be no stable hydrogen, no long-lived stars, no hydrogen containing compounds
If the expansion rate of the Universe were larger, no galaxies would form
If the expansion rate of the universe were smaller, the universe would collapse even before starts formed
If the mass density of the Universe were larger,  overabundance of deuterium from the big bang would cause stars to burn rapidly, too rapidly for life to form
If the mass density were smaller, insufficient helium from the big bang would result in a shortage of heavy elements
I could go on, but lets cut to the chase. 
How do you account for this:
(A) Inflation
(B) Multiverse
(C) Top Down Cosmology
(D) God/Designer
(E) Some new innovation you're working on or have that can win a Nobel Prize in Physics
Let's be clear on the options.  Either we have some law that forces the fine tuning, which does not exist unless you know otherwise, or we put it to chance, which is obvious nonsense given the math, or it's a designer.  Unless you have a fourth option? 

A:

Why questions ultimately in physics have the answer "because". 
What physics does is to model mathematically from existing observations  the observed universe, in an ideal case a theory of everything towards which physics aims.  The models are validated by predictions for new unrecorded at the time the  model appeared, phenomena.   The mathematics of the model, any theoretical model, tells us how from a certain level of observation a next level can be predicted and the model  validated by new data.  
As we go up in the mathematical justification how something comes to be, we end up with the axioms/postulates on which the theoretical model is based, and then the answer, why these axioms, becomes because that is the way the data validates the model.  A tautology, but that is what physics is about.  After this one goes to metaphysics or religion. 
Let me give you an example:  The question Why the hydrogen atom is stable and has the specific spectrum is answered by the quantum mechanical mathematical model of the hydrogen atom, of how given the constants the solutions fit perfectly the observations ( spectra etc). 
Now in this example, before the quantum mechanical solution, one could ask why this spectrum, and the answer would  be ,assuming the Bohr model, these are the numbers that describe the atom according to Bohr postulates.  As with the hierarchies you are discussing in the question, where, the answer to your Why's  is because  "these are the numbers that describe the observations". 
We then see that as physics progresses the frontier where the why questions  are answered with a mathematical model that pushes the "why" to higher postulates ( for the hydrogen atom the postulates of quantum mechanics) enlarges.  It is not impossible to think that once a Theory of Everything (TOE) model is  found and validated most of these questions in your list will have been pushed to the postulates of setting up the TOE model. 
And then there would again be the circular argument.