Can local experiments determine whether spacetime is static or not? As far as I understand, there is no local experiment that can determine whether one has crossed the event horizon of a black hole or not.
At the same time, spacetime in a black hole is non-static if I understand correctly, and outside the even-horizon it is static.
So the answer to my question is "no", at least in the case of black holes.
Is this a general feature?
What puzzles me is that static spacetime vs. non-static spacetime seems to be quite a far reaching difference for any observer that experiences it, yet it is confusing that no observer should be able to determine that fundamental difference by conducting a local experiment.
 A: 
What puzzles me is that static spacetime vs. non-static spacetime seems to be quite a far reaching difference for any observer that experiences it, yet it is confusing that no observer should be able to determine that fundamental difference by conducting a local experiment.

One can easily define an inertial frame of reference locally in curved spacetime. Thus special relativity holds locally and no experiment can show any effects of gravity including those you mentioned.
For more please see Equivalence principle and local inertial reference frames 
A: The statement that the Schwarzschild space-time is "static" outside of the black hole should rather be stated as

There are particular families of observers in the exterior of the Schwarzschild black hole to whom the space-time seems static.

On the other hand, the statement that the interior is "not static" should be stated as

There is no family of physical observers in the Schwarzschild interior to whom the space-time seems static.

But notice one thing. The fact that there is some family of observers in the exterior that sees the space-time as static means nothing, really, if you are a generic observer. If you are free-falling into the black hole, I guarantee you your surroundings will look pretty dynamic to you both when you are above and below the horizon. 
The point is, of course, that the geometry seems static to observers at the horizon that are infinitely accelerated. So the notion of the space-time being static necessarily breaks down already above the horizon for any experiment with finite resources. You can then see that the operational definition of a horizon will always be somewhat "fuzzy".
A: 
Can local experiments determine whether spacetime is static or not?

Yes, if we allow that “local” observations include the ability to calculate local invariants constructed from curvature tensor and its derivatives. An experimenter then can locally determine whether spacetime is (locally) static or not. This could be achieved by calculating appropriate (finite) number of curvature invariants, and finding if there is a timelike direction along which their Lie derivative is zero.
The conflict between the first and second paragraph of the question comes from mixing properties of “general” black hole spacetime and specific static/stationary black hole solutions.

As far as I understand, there is no local experiment that can determine whether one has crossed the event horizon of a black hole or not.

That statement is true for a “general” black hole spacetime about which no apriori information is given to the observer. However, if observer has some “external” information about the spacetime where she is performing experiments, then  under certain conditions it would be possible to determine the moment of the horizon crossing.

At the same time, spacetime in a black hole is non-static if I understand correctly, and outside the even-horizon it is static.

“General” black hole spacetimes are neither static nor stationary outside event horizons. Schwarzschild metric is a specific example of static black hole solution, it has a Killing vector field (KVF) that is timelike outside the event horizon while inside the horizon there are no timelike KVF. So, for Schwarzschild black holes this statement is indeed true. But “realistic” black holes cannot be truly static or even stationary, they absorb mass and exist within evolving universe. 
Nevertheless, once black hole stabilizes we can say that certain finite region of spacetime around it is approximately stationary (if black hole is rotating) or even approximately static (if nonrotating). Perturbations within this region will decay exponentially with a characteristic timescale of Schwarzschild crossing time $r_s/c$, so if there are no continuous external sources of perturbation, black holes would become stationary quite fast (wrt ordinary timescales) and with a high degree of precision.
For static and stationary black holes an observes can locally determine the moment she crosses event horizon by measuring certain curvature invariants. For example, for the Schwarzschild metric, invariant combination $R^{αβγδ;\epsilon}R_{αβγδ;\epsilon}$ crosses zero and switches sign as one crosses the horizon. This paper outlines the Gedanken-experiment that could be used to locally measure the invariant and thus detect the horizon. This paper provides invariant polynomial for Kerr metric, and this paper generalizes the technique to an arbitrary stationary black hole spacetime.
If the black hole is not stationary, calculation of such invariants could still be used to locate an approximate position of the event horizon. For a black hole that is stabilized the difference between calculated and actual horizon could be exponentially small (with a factor $e^{-  A \tau c /r_s}$, where $\tau$ is a time after or until measurable perturbation and $A$ a constant of order unity).
