das
Regeneration In Process
-Is that capital Würzburg?
Fulda seems more like it.
-The rulers are of the House of Leiningen, correct?
Yes.
-When'd they put in the Kiel Canal, and why?
1864, for the sake of easier transit. Also, in avoidance of crazy Danes.
-Would you mind doing a brief recap of the HRE's history (emphasis on political natch) from the divergence? (I know some of the basics, mostly centering on Alexander's war of conquest, and I remember stuff from the two question sessions in the Alternate History threads, but much of the earlier part of the second millennium is still blank.) Sort of like what you did with Azale and Chernigov.
EDIT: Forgot to add something in here.

The HRE is much harder to summarise than Chernigov. Still, I guess I could do a really basic rundown.
There were no Hohenstauffens and the HRE began to disintegrate earlier but slower and less spectacularly. The Imperial crown in the 13th-15th centuries passed back and forth between different dynasties; some major ones, like the Wittelsbachs or the von Hollands, still emerged to gain great power and prestige, but none managed to consolidate their power properly, and all were prone to infighting, as was the Empire itself. Meanwhile, from the 14th century on, the religiously-coloured wars with France made life more and more difficult for the HRE's lesser principalities. The southwestern statelets eventually formed the League of Stuttgart to counter the French; the House of Leiningen emerged as the leader of the League, and has used its subsequent martial successes for political leverage. In the early 16th century, the Leiningens finally gained the throne; at first they didn't seem much more tenacious than their predecessors, but they were able to use dynastic marriages and political manipulation to build up an impressive power base and keep everyone else either irrelevant or on their side. It took some doing and a serious civil war, but by the 17th century the Holy Roman Empire was hereditary, recentralised and ready to fight various Slavs, Italians, Hungarians and Egyptians in addition to their usual repertoire of Frenchmen and Danes (the latter having used the earlier turmoil to expand into northern Germany before getting thrown out of there in the 16th century). The rest you probably already know.
-What's the HRE's political structure, i.e. how decentralized (anything like the old Circles?), is the succession officially hereditary or elective, that sort of thing.
It is actually rather similar to the Second Reich, though slightly less centralised. And the succession is hereditary.
-What cultural minorities exist within the HRE, where, what religions do they follow, and how well have they been assimilated?
Firstly, the Jews (who are present in considerable numbers (largest concentration in the world, and bigger than in OTL due to refugees from France and such) and are only partly assimilated) are practicing various kinds of Judaism, but virtually everyone is Roman Catholic with differing degrees of devotion and often fairly divergent beliefs and practices. Apart from some Italian and Polish minorities and the semi-assimilated Czechs, most of the population is what is called German (for a somewhat broader definition than in OTL); note, however, that there is some extreme cultural regionalism, especially in Bavaria, Frisia, Alsace and Prussia. And ofcourse there are assorted natives in colonies, including Livonia; degrees of assimilation may vary wildly.
-Are there any irredentist claims that the HRE has or particular rivalries with other powers?
Not that much in the way of irredentism, though possibly some with regards to Italy. As to particular rivalries, France is the rival of anyone within reach, Denmark is incredibly annoying and the HRE fought Tver for centuries over the eastern lands.
-How did the colonization of the Orinoco territory (and Belize, I guess, but that's not as important) happen - how did the HRE acquire it, what are the social strata in the colony, how profitable is it, and how stable is the political situation there (any independence movements or native problems?)?
Back in the 16th century, the Imperial merchants first established themselves in the area (and in some other places too). It was only in the late 17th-early 18th centuries that these outposts and lesser colonies were turned into a proper Imperial Colony, expanding into the inlands. The population is mostly German and Jewish; natives are mostly insignificant by now. It is mostly an average settler colony, and is moderately profitable. For now the population is mostly content with the considerable self-government it has been granted thus far.
-What major economic concerns does the HRE have in other countries, what countries, and of what import to the economy?
It has a considerable economic presence in Poland, Hungary, Croatia and Serbo-Romea, and less in various other areas, such as foreign colonies; and trade is ofcourse of great import for the economy, though not as much as one might expect. Also, there was a major recent economical crisis, so...
-What are the status of those Hungarian, Illyrian, and Polish states and their relations to the HRE - puppets or something?
Poland was traditionally a puppet or even a part of the HRE; it has relatively recently broken away and has been trying to retain its independence, though ultimately backing down from its attempts to make an alliance with Tver. The status is uncertain.
Hungary has an unpopular Imperial puppet government.
Croatia is independent, but definitely pro-Imperial in allegiance and foreign policy.
-Why didn't the Swiss ever pop up (or if they did, why were they reabsorbed into the HRE)?
The Forest Cantons never did rise up because they never lost their privileges in the first place (well, subsequently all those things had gone through a revision, but that was later). They remained just another part of the HRE, as far as we are concerned.
-How well militarized is the HRE (i.e. size of army per capita and how well it has performed in recent years), how modern is their amry, and who controls it within the political structure of the empire?
The HRE has a very large and well-trained, mostly modern army, controlled by the Holy Roman Emperor and his General Staff.
-How about their navy - has it been fairly well developed and modernized, what is its size, and who has control of the navy?
The navy is pretty middling in most regards, though it is relatively modern. It is under the same general military command, though it is not integrated with the army.
-Are the colonies and outposts in Africa and South America primarily military outposts for use by the navy, trade posts, or some other form of colony (see the previous discussion on colonies in a certain other thread)?
Orinoco aside, colonies are trade outposts and coaling stations.
-What events have happened most recently in the HRE - any political shocks or anarchism or conflicts with other countries?
A major "shock", to say the least, has occurred some three-two decades ago, when, after the death of the Kaiser, a series of nationalist uprisings led to the independence of Poland and Italy despite the Imperial military's best efforts. Subsequent reform efforts were undermined and partly derailed by the growth of a worker's movement, which was partly thwarted by social reforms.
-Do socialist or communist (or Egalitist) groups exist within the HRE, and how are they treated by the government and the populace at large?
Arbeitist, yes. Their influence is waning, and the more radical ones are illegal; others are grudgingly tolerated.