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?
 A: There is a reasonably well-developed physics/statistics literature on this topic
There is actually a fairly well-developed literature on this matter in physics and statistics journals.  Some good papers on this topic are Zhang-Yuan and Bin (1985), Keller (1986), Vulović and Prange (1986), Gelman and Nolan (2002),
Mizuguchi and Suwashita (2006), Diaconis, Holmes and Montgomery (2007) and Strzałko, Grabski, Stefański, Perlikowski, Kapitaniak (2008).  The general method in the literature on this topic is to develop physics models of the coin-tossing process (under idealised conditions) and examine the sensitivity of the outcome to initial conditions such as the coin velocity, angular velocity, etc.  A coin-toss is generally highly sensitive to initial conditions, which means that even slight random variation in these initial conditions will lead to a probability that is extremely close to "fair" for the outcome of the toss.  The literature involves a combination of physics models and empirical tests, some of which are done with mechanical coin-flipping mechanisms and some of which are done by humans.
The details of the results in the literature are somewhat complicated, but the general theme that emerges is that the mechanics of a coin-toss is so sensitive to small changes in initial conditions that it is quite "chaotic" and so the uniformity assumption (i.e., equal chances of heads/tails) is reasonable in practice.  In particular, if the coin is allowed to bounce after the flips, before coming to rest, this introduces extreme sensitivity to initial conditions.  Diaconis, Holmes and Montgomery (2007) show that you can get results that depart a bit from uniformity under extremely idealised conditions (no air resistance, no bouncing of the coin, high consistency of initial angular velocity, etc.), but this is not reflective of most practical cases of coin-flips by humans.  Strzałko et al (2008, pp. 90-91) conclude the following:

In practice although heads and tails boundaries are smooth the distance of a typical initial condition from a basin boundary is so small that practically any finite uncertainty in initial conditions can lead to the uncertainty of the result
of tossing. This is especially visible in the case of the coin bouncing on the floor, when with the increase of the number of impacts the basin boundaries become more complicated. In this case one can consider the tossing of a coin as an
approximately random process.

If you have a good look at the literature on this topic you will see that you can obtain a coin toss that is extremely close to "fair" by tossing with a sufficiently large angular velocity to ensures that the coin will spin a reasonable number of times before landing.  You can also get closer to fairness by allowing the coin to bounce on a hard surface instead of catching it.  And of course, if you pick up the coin for your flip in a manner where you pay no attention to the initial face-up side, then the initial condition is arguably already uniform, which would imply uniformity of the outcome.  Good luck with improving your coin-tossing technique!
A: 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.
