Legend of Galactic Heroes/WMG

The peculiar aestethic of the series is caused by a visual Translation Convention.
As noted elsewhere, it is exceedingly unlikely that fashions and architectural styles from the 80s (whether the 1980s or the 1780s) will be in vogue at the end of the 36th century. Also, after 1500 years modern languages that do not go outright extinct along the way are likely to have evolved into something unrecognisable to contemporary ears and eyes. The look of the series is likely intended as a visual shorthand to suggest the character and outlook of the respective society - the Empire is authoritarian, autocratic and militaristic, therefore it looks like Prussia or Imperial Germany. They likely emphasise cultural uniformity, therefore everyone has a "German" name even though it is unlikely that every one of them had "German" ancestors - those who didn't have them to begin with were encouraged or forced to assume such names. The Alliance is supposed to reflect contemporary (for the 1980s) liberal democracies, therefore they look like they do. Their use of Engrish is likely intended to suggest that they are similar in outlook to the "Anglo-saxon" societies of our world, with an emphasis on free enterprise (complete with hints of plutocracy) and a consumerist society. Perhaps the Empire actually looks more like the ones in Warhammer 40000 or Dune, and the Alliance is like the Federation in Star Trek.

The Galactic Empire marked the bittersweet return of Prussia.
Given how blatantly Prussian, or at least German, the Empire was modeled, it wouldn't be surprising if a fading, near-dying Gilbert suggested the idea to Rudolf in the first place. Hiding among the military brass, he participates in the Kaiser's campaigns (at the cost of millions of lives). As time passes, he once more becomes rejuvenated, just as he had been in the distant past. Unfortunately, Hungary/Elizaveta had been a victim of World War 3, by then ancient history; it doesn't help that Hungarians apparently are extinct by the 36th Century. Meanwhile, Austria, or someone claiming "descent" from Roderich, is lurking in the shadows, undermining the Reich with the help of Reinhard.
 * Come to think of it, if this is the case, it mirrors in a way coronation. Gilbert would be the premier Nation in the galaxy, and by technicality, the only one. But the ones he'd want to see, Hungary and Austria, not to mention his brother, would have long faded into the dustbin of history.

Reinhard has an allergic reaction to vaginas.

 * Hardly, the disease was foreshadowed with him occasionally laid low by inexplicable fevers long before that.
 * Proximity/exposure to Hilde, perhaps?

Job Trunicht was aiming to unite humanity under a constitutional monarchy the whole time.
Knowing that it was only a matter of time until the Empire won, especially after, Job Trunicht knew that the best way to preserve the future of democracy was not to fight the Empire and make themselves unsympathetic enemies, but to willingly capitulate to the Empire's terms and go about establishing constitutionalism from the inside.
 * He explicitly states on-screen that he will support whichever political system that benefits him personally. He probably would have supported constitutionalism if it would gain him more power and influence.