That may be true for the southern border in the map, but in the North the left half of the "maximal extent" seems to be from before the Austro-Prussian war of 1867, and I'm not sure when they had the right half, but that territory was occupied by Germany in WWI rather than Austria-Hungary.

The paper is here: http://ftp.iza.org/dp5584.pdf

Some quotes relevant to the questions you asked:

To ensure that the “control group” to which the Habsburg “treatment group” is compared does not include locations that had actually also been exposed to the Habsburg treatment at some time, throughout the definition of the Habsburg variable H is an indicator of whether a location has ever been part of the Habsburg Empire. By contrast, defining the Habsburg variable by Habsburg affiliation at any particular point in time would mean that part of the “control group” had also received a “Habsburg treatment” at some point in history. For example, defining the Habsburg treatment by its borders just before its demise in 1918 would mean that several regions that had been part of Habsburg until 1908 would constitute a substantial part of the control group in the border sample. However, as one robustness analysis below, we show that results are robust in a specification that restricts the analysis to locations that were part of the Habsburg Empire in 1900 (and their control locations) while dropping all locations from the analysis that had been part of the Habsburg Empire at some time but were no longer in 1900.

The specification of our basic model assumes that the Habsburg treatment effect is independent of the duration of treatment. To test for the validity of this assumption, we will also estimate models that allow the Habsburg effect to differ by length of a community’s affiliation with the Habsburg Empire: […]

— Empirical Model, p. 13

Third, and most importantly, in contrast to the basic assumption of a regression discontinuity design, there are a number of reasons to expect some sort of diffusion to and interdependence between neighboring towns across the former Habsburg border. Such reasons include migration and marriage between neighboring towns; local spillovers whereby inhabitants of an Ottoman town just across the border observe that well-functioning public services may have positive consequences, so that they may imitate them to some extent; “frontier” effects in that authorities may behave differently if located close to the enemy, such as when a Habsburg local authority may behave more authoritarian if the Ottomans are next door; and political competition across neighboring locations. All of these effects would mean that the Habsburg status of one town may affect outcomes in towns that are located directly on the other side of the border. Note that any of such diffusion or interaction effects work against our identification strategy finding a significant difference between Habsburg and non-Habsburg locations in the border sample.

— Empirical Model, p. 15

(There is also some relevant discussion with regard to your questions in the Results section.)

Note that all of the things you mentioned would have made it less likely to find an effect.

This looks like solid work. Like always (especially when social sciences are involved), there are problematic aspects to it. Social science is damn hard to get right.

Thank you for the link to the paper. I wasn't able to find in the paper any basic data, only derived coefficients. The map is the same, including the "maximal extent" which happened only at the few years the territories were occupied during WWI. The big movements of big amount of people over the territories during the turbulent history are also somehow not addressed. There's a unique history behind each settlement, not taking that in the consideration or not showing what you actually did I believe any conclusion can be generated. I still question some premises and the execution of the paper.