This is a good question, but it suffers a bit from backward projection. "States", as we know them today (constituted nation states) are a fairly modern (as in early modern) idea. Before 1648-9 (treaties of Munster and Westphalia, ending the thirty years war and giving birth to "modern" international law), the feudal understanding of sovereignty was deeply tied to inherited rights of the feudal ruler, dynasties and inherited traditions (some of which, sometimes were "made up" to justify annexations and wars).
In the course of history of the Holy Roman Empire, settlements and regions of all kinds had specific forms of ownership (Duchies, Bishoprics, Trading City Republics etc). Among these, self-governed communities were becoming more and more common over time, and there was a large variety of factors in this development involved (cf. Peter Blickle, Kommunalismus, https://academic.oup.com/ahr/article-abstract/109/1/268/108903?redirectedFrom=fulltext). In short, this is a really complex topic, but to break it down to a few important acpects:
- the Emperors had little direct rights over his subjects, unlike the Emperors of Imperialism in the 19th century; but among these rights was his ability to declare a city (or settlement) reichsunmittelbar (roughly translatable as "under direct Imperial sovereignty"), which gave such a city republic the right to claim that they could not be subsumed by any other feudal lord under the Emperor. Thus, an important Bishop or Duke could not hope to gain acceptance as a ruler over such a city even if conquered. As Emperors were constantly short on money on their military campaigns (especially in the "high middle ages" 10th-13th cty., e.g. under rulers like Friedrich I "Barbarossa"), they effectively sold such rights to independent suzerainty for good payment to an ever increasing number of cities. By the late 15th century, the Holy Roman Empire counted over 1,200 such self-governing city republics, often in a form like Venice -- not truly modern democracies, but often definitely small city republics with regular elections, councils etc.

- On the other hand, the countryside was undergoing a process of increasing centralization, at least in the sense that more and more aristocratic families tried to gain larger contiguous swathes of land. Since, however, international law did not exist in its current form, the death of any ruler and varying laws of inheritance and succession, could muddle the landscape further. An aristocratic family would often have ownership of various disparate lands, villages and towns all over the map, and would "feud" often enough over rights of access, control or ownership of various lands as claimed through previous family ties or letters.
There are many examples for both of the above factors in the long course of the 11th century until 1806 (end of the HRE).

Now, for your two specific subquestions:

- Were people allowed to travel freely between them? More or less, yes. But there were customs, paperwork, letters required for most things more complicated than a few miles travel. This is hardly imaginable today, but this was simply daily life in this long run of history, and was second nature to any merchantman of the time. A hundred mile voyage along the Rhine river, a main trading artery, could result in as many as 34 custom checks in some phases of this history. The split into a thousand sovereignties was Germany's historic trauma (hence the nationalist slogan in the 19th century that "unity" came before "justice" and "freedom" -- the latter were obviously unthinkable without the former, which explains the strange word order in the German national anthem, "Einigkeit und Recht und Freiheit" ("Unity, Justice and Freedom for the German Fatherland"). Unity, in this case, does not mean conformity, but a standardized system of law guaranteeing your ability to travel freely without hindrance. Many a drama was written about this aspect of daily life in the Empire, most famous possibly Heinrich von Kleist's Michael Kohlhaas. (Cf. https://en.wikipedia.org/wiki/Michael_Kohlhaas)

- The right to mint coins was with each sovereignty. While some currencies were widely considered more favorable than others, we are talking about a time where "fiat" money (i.e. money purely based on trust, rather than Gold or Silver) was far less common. Nevertheless certain currencies were considered more "reliable" than others (quality of coins and ubiquity of their acceptance in trading arrangements). The most famous currency in the HRE was without doubt the 16th century Bohemian ”Joachimsthaler" silver coins (cf. https://en.wikipedia.org/wiki/Tolar) , colloquially just called the "Thaler" or "Taler" (Czech: Tolar), the Saxon soft pronounciation of which sounds like "Dollar", which is also the true origin for America's currency designation, the Dollar. However, all currencies were floating freely around, and traders and merchants were well versed in dealing with literally hundreds of different coin types until the modern age brought about the standardizations of modern currencies for large nation states. A tiny region in the HRE like the Swiss Confederacy, itself consisting of dozens of states, had individual coins for each Kanton (State) well into the mid 19th century: A Swiss citizen of 1840 was still used to deal in their daily lives with over one hundred different coins - and those were just the Swiss ones!

Hope this helps a little.