There were no real borders for the longest time, only zones of influence where nobles could exert their authority (which is moot if nobody cares about the person) and lords above those nobles, to whom which the nobles would swear loyalty to (most of the time).

These zones are very fluid. Within a mile of your castle? That land is definitely under your control (just ask the last guy who tried to claim that land). Twenty? Eh, there's an army there that's as big as ours. Let's just let them sit there for a while. What you have instead of a fine border is overlapping influence; some of these regions will have undoubtedly survived into the era where borders aren't so fluid.

In other words, the borders will vary depending on who you're asking and when. These borders survived into the era of map because of how much easier it is to maintain the status quo where regional powers can do their thing in exchange for supporting the powers that be. It's more trouble than it's worth to assign borders (read: conquer) to nations/people who only have their interests at heart. The Holy Roman Empire was an empire in name only.

This is the extent of my knowledge though. Others will have to come by and clear things up or correct my mistakes.

Keep in mind that what appear to be "crazy borders" on a map may be completely sensible on the ground. The modern idea of borders being straight lines and thinking of borders as absolutes is something that happened as mapping technologies improved and people started thinking of countries in terms of topographical area.

In the past borders could be very sensible locally (following ridges, rivers, streams, edges of forest etc) but look less so when viewed from above.

When you couple this with European methods of title inheritance, it leads to things that look like the HRE did.

I'd be interested as well in knowing specifically how this affected merchants and trade. Was there a heavy amount of "red tape" to do business in the HRE given all of the different entities?

For the HRR, this map is rather below par in fragmentation. A hundred years before, it was worse, two hundred much more so, and in 1400 it looked like this. We can't see the smallest units on these maps, of which there tended to be a lot.

This list tries to list the states which were on the territory of the HRE, but it list some doubly (Nürnberg was a county and then a free city, for example), and there were quite certain a lot more, which were not in the Reichsstände (estates of the realm), so a lot of Reichsritter (Reichsknights, who sometimes only comanded a village or less - presumedly the most famous of which is Götz von Berlichingen - which were directly under the authority of the Kaiser) and some small abbies and such are not listed.

There are more than 500 states on that list.

How did that come to pass? Maybe the main reason was the traditional germanic division of the estate after the death of people (think à la King Lear, or Charlemagne). However, there were some countermeasures against that. Namely, there were a lot of fights between brothers and relatives. Bavaria, which seems to be quite a closed territory in both maps, was divided multiple times following the death of rulers, and after a few wars and after a few branches died out, reunited in 1505 after that, the succession laws were changed.

Another reason was sometimes the eroding of the central authority. When for example the Hohenstaufen died out - they were dukes of Swabia and emperors - there were several contenders to the dukedom. After several wars, nobody could establish himself as duke, and the land stayed fragmented, some former vassals of the duke established their own independent states (a lot of them Reichsritter).

Another reason was ominous music the dead hand. The church as such was never dying. So, given that some people would donate land, the church was always accumulating that land, never have to divide it, never have to pay the normal taxes on the death of it's regent. In short, donated land was lost to the church forever in the eyes of everyone else. This lead to huge accumulations of land by some bishoprics and abbies, of which some were made independent states.

To "How often did boarders change within the empire?": All the time. It constantly happened that some dynasty or the other died out or that some states were leading war against each other, on the slightest of injuries taking bits of land from each other. For example, one of the succession disputes nearly launched into an European war in 1610, when the ruler of Jülich-Cleve-Berg died and the catholic league and the emperor supported one heir and the protestantic union, England (and France, before Henry IV died) and the United Netherlands supported another.

Austria started as a March and over time married into other families, inheriting all that territory.

Silesia was a part of the Bohemian crownlands, but Brandenburg was able wrench concessions out of Austria during a disputed succession between Bavaria and Austria which caused a succession war. Brandenburg then later seized the rest by force. Also, if someone not a member of the Austrian line were to be elected to the Bohemian crown that generally meant an Imperial Ban on the person. An Imperial Ban being basically 'This person has no rights and may be killed freely"

The red dots look like Imperial Free Cities, which answered only to the Emperor with no intermediate lord. These come about through various means, such as the intermediate liege line dying out or being granted to the catholic church and not passing back in. In the later case a lot of contention could happen in the imperial court of church vs free zone admins wanting autonomy.

Despite supposed to being part of the same Empire, Imperial Princes would often go to war with each other. I do believe this was made fun of in Candide. Although I think this map depicts a time before Voltaire.

If you look carefully in the Wurtenburg area you can see a faint outline of the former Duchy of Swabia and how it fell apart.

I'm not a historian but that is what I know on the subject.

Smaller and smaller land divisions in the HRE came about by Salic patrimony. Basically all male heirs are entitled to a piece of the family's holdings and then their heirs and so on. Carry this out over a number of generations and you get the fractured messes of the Palatinate and the various Hesse-XXXX Landgraves.

Edit: autocorrect

I was told by a Liechtensteiner host that Liechtenstein was one of these kingdoms that never assimilated, but the only thing on this map that seems to match modern Liechtenstein borders geographically is Fürstentum Isenburg. Any insights?

What you need to consider is that loads of European countries started off like this, with small areas under the control of various nobles (think of the time following the collapse of the Western Roman Empire). The thing is, though, that while countries like France and England developed into centralised states pretty quickly, the HRE never did and thus the borders (which aren’t unnatural, most of them would follow rivers, mountain ridges etc) stayed put. If you’re interested in the reasons for this process never happening to the HRE, have a look at this thread where I wrote some longer answers.