Why does Fumi's master look so suspiciously like Mashiro?
Because that was the FIRST Mashiro, the Crystal Princess of Mai-HIME. They were trying to tie the continuity together, where Fumi in Mai-HIME was also Mashiro's servant. This Mashiro was named after the original Crystal Princess, and the whole series is basically an exercise in Identical however-many-generations-removed-grandchildren.
Which opens ANOTHER can of worms, considering the first Mashiro departed through the Door to Valhalla at the end of Mai-HiME.
Well, first things first. The "Mashiro" we see in Mai-Otome is in fact not the real Princess Mashiro (that's Nina), and even if the two series were occurring in the same continuity (which was never explicitly confirmed), even the flashback with Fumi is well into the future compared to the present-day Mai-HiME. So if the series are indeed related, perhaps that was the descendant of the original Mashiro, or simply a member of the uninterrupted royal bloodline of Windbloom who just happened to look like Mashiro.
Why does the scenery look more "modern" than in Mai-Otome itself, even though this is a prequel?
The entire world's Schizo Tech doesn't make it that strange. Also, architecture fasions go through nostalgia periods regularly so old fashioned looking buildings could easily be newer. There's also the fact that the buildings in Windbloom could be more than 20 years old while the hotel they visit in this OVA could be brand new. If this is the future, why would they build gothic castles at all if there wasn't Rule of Cool going on?
Maybe its just Windbloom with the nostalgia kick or just old architechture from an earlier age much like Vienna or Rome in the modern age. Sifr isn't set in Windbloom and perhaps this ultra-modern look is favoured elsewhere. Granted Windbloom is the most technologically advanced city but its implied that Windbloom has been around since the foundation of Garderobe since the Harmonium is situated underneath Windbloom Castle.