What *physical conditions* give rise to "fairness" of a coin toss in statistics? The majority of us are aware of properties related to fairness of coins or dice and all standard text/reference books cover the definition of fairness of a coin.

In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin.

But in fact, I have never read about fairness of the toss ---i.e., do I have to toss a fair coin vertically in air with same amount of force applied on coin each time or can I toss a fair coin with any amount of force ---banging it on multiple surfaces--- before it finally lands on flat surface at rest position each time?
The word "toss" is loosely defined in statistics, in fact, in the Cambridge dictionary of Statistics there is no definition of a toss.   The general dictionary indicates the definition to be:

move or cause to move from side to side or back and forth

Same goes for rolling of a die.
Is there anything called fairness of toss?

I think that the "fairness of the toss" is usually implied by "fairness of the coin." In other words, it is assumed that the toss will be essentially the same and will lead to an outcome that is only determined by the fairness of the coin.
Ie, suppose that the "toss" consists of dropping the coin from one inch onto a table.  If heads is up when the coin is dropped, heads will be up when the coin lands and vice versa for tails.  Hence, even if the coin is fair (when tossed properly, it will land heads with prob .5) the experiment will not be.
The "fairness of the coin" however is usually just shorthand for "fairness of the coin and toss and other conditions of the experiment," such that the scenario above with the drop from one inch is preempted.
I think this is an important distinction, though, and a good question on your part.  Assuming the probability model under fairness of the coin implies fairness of the toss is kind of a fallacy.  It would be an error in the scenario described above with the toss from one inch, and maybe in other scenarios that you are imagining.
In other settings, we often consider maybe just one piece of the picture when modeling random events, and we are badly mislead in our ultimate conclusions for doing so.
