How can we be sure about the constancy of atomic clocks as in the Hafele and Keating time dilation test? Atomic clocks were used in Hafele and Keatings experiment which supposedly helped to prove time dilation.
Time Dilation Proof - Hafele and Keating
How can we be sure other forces didn't act upon the clocks as a result of their being at different speeds and elevations? Perhaps the elapsed time may have been the same for all and only appeared different as a result of the clock's atoms behaving differently in different environments.
How might the presence of dark matter (presumably all over the place) impacted the clocks if say the dark matter is stationary relative to the earth's rotation?
What experiments have been done to test the constancy of clocks at different speeds and elevations?
 A: There is always a possibility, in any experiment, that some unknown effect was the main cause for the obtained measurements. This can never be ruled out.
The reasonable way to approach this is to first ask what is our null hypothesis. In this case we would like to test general relativity, so our null hypothesis is that there is no relativity and that Newtonian mechanics -- the previous successful theory -- will be able to explain the experiment. The alternative hypothesis, which is the one we would like to test, is that general relativity is correct.
So now we have two predictions for the shift in the clocks. For this to be interesting the predictions should be very different, compared with the experimental sensitivity. We carry out the experiment, and if the result 'agrees' (in a statistical sense that I will not go into) with general relativity, then this is evidence in favor of general relativity. If the results 'agree' with Newton, this is evidence against general relativity. If the results 'disagree' with both, it is evidence against both theories.
Either way, the experiment does not prove that GR is correct or incorrect, for example because of a possible unknown effect. It can only increase or decrease our confidence in GR, namely it can change the probability that we humans assign to the statement 'GR is correct', and even that is only within the regime of parameters where we can carry out experiments. The best we can hope for in science is to have theories that we are very confident about.
A: Newton's predictions are consistent with Einstein's under the appropriate situations.  No thwacking.  You don't get to adjust a theory for every particular experiment, if your theory is well-constructed.  General Relativity starts with a very small number of assumptions, and then creates predictions.  These predictions are bourne out by a very large number of experiments.  From the time dilation of decaying particles, to the laser ranging of the moon, to gravity probe b, to hartle hawking, to the Eötvos experiment.  The effects of time dilation, and general relativity in general, have been measured under too wide a variety of circumstances for concerns about atomic clocks to be a credibe reason for doubt:
http://relativity.livingreviews.org/Articles/lrr-2006-3/
Come up with a simpler, more straightforward theory that does better.  If you can, a certain nobel prize awaits you.
A: If the effect wasn't due to time dilation, then why would the measured results agree with predictions of relativity so well?  On the 25th anniversary of the H-K experiment, the National Physical Laboratory reproduced the experiment.  For the clock flying from London to Washington, the relativistic prediction was that the clock would gain 39.8 nanoseconds.  The measured result was a gain of 39 $ \pm$ 2 nanoseconds.  In 2010, the experiment was done by the NPL once again - this time, around the world.  It went from London, to Los Angeles, to Aucklund, to Hong Kong, and then back to London.  They measured a gain of 230 $\pm$ 20 nanoseconds, compared to relativity's prediction of 246 $\pm$ 3 nanoseconds.  
So, the odds of the clocks being off by just the right amount to account perfectly for relativity are incredibly slim.  
In terms of the accuracy of clocks, I don't know if they have been tested in different environments.  However, NPL's clocks have been tested to be the most accurate in the world.  
A: 
How can we be sure about the constancy of atomic clocks as in the Hafele and Keating time dilation test?

Of any one clock $\mathfrak A$ it can be measured whether its average rates were equal to each other, or by how much they differed from each other:
In regard to

*

*any two (distinct) indications $A_{\circ F}, A_{\circ G} \in \mathcal A$ of an identifiable material participant $A$, with assigned readings $t_{\mathfrak A}[ \, A_{\circ F} \, ], t_{\mathfrak A}[ \, A_{\circ G} \, ] \in \mathbb R$ and


*any two (distinct) indications $A_{\circ J}, A_{\circ K} \in \mathcal A$ of an identifiable material participant $A$, with assigned readings $t_{\mathfrak A}[ \, A_{\circ J} \, ], t_{\mathfrak A}[ \, A_{\circ K} \, ] \in \mathbb R$
clock $\mathfrak A \equiv (\mathcal A, t_{\mathfrak A})$ had run from $A_{\circ F}$ until $A_{\circ G}$ and from $A_{\circ J}$ until $A_{\circ K}$ at equal average rates if
$$(t_{\mathfrak A}[ \, A_{\circ K} \, ] - t_{\mathfrak A}[ \, A_{\circ J} \, ]) = (t_{\mathfrak A}[ \, A_{\circ G} \, ] - t_{\mathfrak A}[ \, A_{\circ F} \, ]) \, \left( \frac{\tau^A_{[ \, \circ J, \, \circ K \, ]}}{\tau^A_{[ \, \circ F, \, \circ G \, ]}} \right), $$
where $\large \left( \frac{\tau^A_{[ \, \circ J, \, \circ K \, ]}}{\tau^A_{[ \, \circ F, \, \circ G \, ]}} \right)$, i.e. the value of the ratio between $A$'s duration from its indication $A_{\circ J}$ until its indication $A_{\circ K}$ and $A$'s duration from its indication $A_{\circ F}$ until its indication $A_{\circ G}$ can and must be measured beforehand (by the familiar methods of the theory of relativity), of course.
If all average rates of clock $\mathfrak A$ between all (relevant) pairs of its indications were equal, then $\mathfrak A$ is said to have run at constant rate.
Also of interest is the comparison between the average rates of different clocks:
In regard to

*

*any two (distinct) indications $A_{\circ J}, A_{\circ K} \in \mathcal A$ of an identifiable material participant $A$, with assigned readings $t_{\mathfrak A}[ \, A_{\circ J} \, ], t_{\mathfrak A}[ \, A_{\circ K} \, ] \in \mathbb R$ and


*any two (distinct) indications $B_{\circ P}, B_{\circ Q} \in \mathcal B$ of an identifiable material participant $B$, with assigned readings $t_{\mathfrak B}[ \, B_{\circ P} \, ], t_{\mathfrak B}[ \, B_{\circ Q} \, ] \in \mathbb R$
the two clocks $\mathfrak A \equiv (\mathcal A, t_{\mathfrak A})$ and $\mathfrak B \equiv (\mathcal B, t_{\mathfrak B})$ had run at equal average rates if
$$(t_{\mathfrak B}[ \, B_{\circ Q} \, ] - t_{\mathfrak B}[ \, B_{\circ P} \, ]) = (t_{\mathfrak A}[ \, A_{\circ K} \, ] - t_{\mathfrak A}[ \, A_{\circ J} \, ]) \, \left( \frac{\tau^B_{[ \, \circ P, \, \circ Q \, ]}}{\tau^A_{[ \, \circ J, \, \circ K \, ]}} \right), $$
where $\large \left( \frac{\tau^B_{[ \, \circ P, \, \circ Q \, ]}}{\tau^A_{[ \, \circ J, \, \circ K \, ]}} \right)$, i.e. the value of the ratio between $B$'s duration from its indication $B_{\circ P}$ until its indication $B_{\circ Q}$ and $A$'s duration from its indication $A_{\circ J}$ until its indication $A_{\circ K}$ can and must be measured beforehand (by the familiar methods of the theory of relativity).
