For your first question, Rigollet has a series of notes(with minor typos) that dicusses basics of this kind of tail bounds. The result you mentioned in the "introduction" is actually the classic Hoeffding bound that mentioned by Rigollet in this set of notes.
High Dimensional Statistics, Philippe Rigollet (2015)
http://www-math.mit.edu/~rigollet/PDFs/RigNotes15.pdf
If you are concerned with the order statistics, you usually want to restrict yourself to a narrower class of distributions, say the classic paper [2].
For a general distribution family, it is almost impossible to obtain a tail-bound on the order statistics due to the concentration of measures phenomenon. Another keyword you may want to look into is "U-statistics" because the (full) order statistic is an example of U-statistics, the following quote from [1] is the best description of the power of U-statistics
Theoretically, for these U-statistics we can study the whole spectrum
of asymptotic problems which were investigated for independent
variables. As a matter of fact, it is necessary to control the nature
of dependence in order to obtain meaningful results. We restrict
ourselves to exchangeable and weakly exchangeable variables, rank
statistics, samplings from finite populations, weakly dependent random
variables, bootstrap-variables, and to order statistics.[1]p.15
Concentration bounds on U-statistics is a research subject that involves many false claims and hardcore techniques, so I think it is not too hard to find one but never too careful to use one.
Update in response to update3.
Why is the maximal uniform spacing so small in magnitude compared to the maximal oscillation in the empirical distribution?
This is not something surprising. you can always have very large spacings but small Kolomogorov-Smirnov norm, which measures on the space of $\mathcal{M}(\mathbb{R})$ while the spacing is measuring $\mathbb{R}$. If you look into Luc's argument, you will see his comments on it.
For the simplest example, given a set of data $\{0,0.1,0.9\cdots,0.9\text{(n repeated 0.9)},1\}$ and $\{0,0.1\cdots,0.1\text{(n repeated 0.1)},0.9,1\}$, their empirical measures has completely different cdfs but they both have the same maximal spacing of $0.8$. as the number $n$ of repeated observation increase, the KS norm between these two cdfs can be arbitrarily close to 1.
Reference
[1]Korolyuk, Vladimir S., and Yu V. Borovskich. Theory of U-statistics. Vol. 273. Springer Science & Business Media, 2013.
[2] Gupta, S. Das, et al. "Inequalities on the probability content of convex regions for elliptically contoured distributions." Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971). Vol. 2.1972.
Best Answer
First I'll address your initial question without taking into account the details of the specific problem. The answer is "yes" if, and only if, the probability distribution is that of a test statistic, where the null hypothesis will be rejected if the test statistic is too big.
When you get into the details of your specific problem you become unclear. When you say "fitted a gamma distribution to some experimental data", it sounds as if you've got a sample and you're estimating the two parameters. Maybe some of the data points will fall above the 99th percentile. If 1% of them do so, that doesn't sound like a reason to reject any null hypothesis that I can think of. If a substantial proportion of them do so, I'd wonder about your method of fitting. If you have a null hypothesis that says something about the values of the parameters, then there's the question of what you're going to use as a test statistic, and then there's the question of what is the probability distribution of that test statistic if the null hypothesis is true. You haven't told us enough about your specific problem to make any guesses about those things.