I suppose 1000 years is a lot to ask about, so let's say 1600 and onwards (yes, I know this includes the Peace of Westphalia, which is a big deal). This map sort of illustrates what's confusing to me. There's the HRE, the boundaries of which seem to be separate from the national borders at times. Some Spanish Territories are in, some out, ditto for the Crown of Bohemia.
-What do those borders mean? What did it mean to be a part of the HRE?
-How was the HRE governed and divided, and did the parts even consider themselves united? Were they?
-If I'm in the Spanish Netherlands, on that map, for instance, who is "in charge", the leader of Spain, or the Emperor?
Well, this is quite the question because the nature of the Holy Roman Empire can seem confusing at first glance (and second, and is just plain confusing). The Holy Roman Empire, was a weird combination between a singular polity, and a collective confederation. The Emperor is the head of state of the Holy Roman Empire, and there was an imperial legislature to pass laws for everyone in the empire (although being as loose a union it was, nothing much ever came from it).
Looking at the Bohemia/Austria/Hungary situation on that map, I can see where the confusion lies. Bohemia, Austria, and Hungary are only held in personal union, so using one colour to represent all three nations is wrong. England, Scotland and Ireland are also all held in personal union, but they are clearly marked as separate countries. All of Bohemia and Austria are in the Holy Roman Empire. Hungary is not. For a modern example, the UK and Canada are held in personal union by Elizabeth II, but the UK is in the European Union, Canada is not.
The red border that surrounds the area is the national border of the Holy Roman Empire. The other borders are the borders of the states within the empire. To be a part of the Holy Roman Empire is a bit similar to being in a part of what NATO is now. It offered protection and alliances to the states within it.
The Empire was divided into many, many, subdivisions. Basically by anyone who could claim land, though most were ruled over as principalities, duchies, or the occasional kingdom. They ruled over with great degrees of autonomy, but all owed fealty to the Emperor (in theory). Truthfully, the Holy Roman Empire was not very unified at all, they were just a collection of all but independent states.
Using you example of the Spanish Netherlands, since you are not in Spain, just a country held in personal union, your head of state is the Holy Roman Emperor. But in reality, all decisions are effectively made by the Spanish monarch. Similar to in the US, the governor and the state legislature takes care of all the day-to-day work, but the federal government and President represent all the states internationally.
TL; DR: It's complicated. Seriously complicated.
-What do those borders mean? What did it mean to be a part of the HRE?
The borders mean that there are some states within the HRE. The HRE during this period of time (and later) was mostly a federation of states, which had the same Emperor and the some mutual laws and institutions. These laws were a strange collection of centuries old privileges for the most part, but sometimes there were tries for reform for more coherent laws (For example: the Constitutio Criminalis Carolina from 1532, was a criminal law for the whole HRE). The institutions were the Reichskammergericht (a law court) the Reichshofrat (which was in essence another law court) and the Reichstag (Imperial Diet). There were also some mutual taxes.
-How was the HRE governed and divided, and did the parts even consider themselves united? Were they?
The territories were ruled by their monarch. This could be the duke of Bavaria, the archbishop of Mainz, the prince-elector of Brandenburg, the Imperial City Augsburg, even the Reichsritter (Imperial knight) zum Stein. To put it short, after centuries of civil wars and a weak state, the princes could mostly do what they wanted. After 1500, the principalities were put together in Reichskreise (Imperial Circles), to better organize the defense, Imperial Diet and such. At first there were six, later ten. However, there were some parts of the HRE not in this Circles. Bohemia, the Old Swiss Confederation and Imperial Italy. Also the territories which had imperial immediacy (they only had the Emperor and institutions over them). There also was an Army of the Empire, but it was quite unimportant most of the time.
There was some degree of unity among the populace and monarchs, mainly through the cultural aspect (maybe they also were proud of their Emperor, for example Maximilian, the duke of Bavaria in the Thirty Years War, was a blend of ambition and devotion to the (position more than the actual man) Emperor), however there are some indications that local identy beat the HRE identity (the Swiss, the Netherlands).
Sometimes they were quite united (the Second Ottoman Siege of Vienna) and sometimes they allied and made war on one another (the Smalkaldic War, the Thirty Years War, countless succession wars).
-If I'm in the Spanish Netherlands, on that map, for instance, who is "in charge", the leader of Spain, or the Emperor?
If you were in the Spanish Netherlands in 1648, the man in charge would be Leopold Wilhelm of Austria, the cousin of Phillip IV of Spain. The kings of Spain would have a gouvernor in the Spanish Netherlands, who would, a lot of the time, like in this case, also be a Habsburg. However, your sovereign, who would be on your coins and in your prayers, would be the King of Spain. But sometimes the Emperor would be in your prayers, because he protects you from the heretics and the Turks.
