It would be quite tempting, there, to exclaim "of course!"; yet there is a dangerous teleological trap to be wary of. As you yourself noticed, it is not because a project or vision of some kind of union has been in the past proposed in Europe (as a continental reality) that it is necessarily an european project, in the sense that it would aim toward something we would today call Europe, be it as an identity or as a set of institutions (the EU). One could argue that since roman times there was a sense of continental unity, and that in the Middle Ages the catholic Church was a de facto cross-European institution: yet confusing the imperium, which encompassed parts of Asia and North Africa, or even the Church, which thoroughly identified as Christian or universal, with something "European", would be but an anachronism. It is quite tempting for someone with even the slightest leanings toward federalism, beside, to retroactively build up a narrative of ongoing european unity leading to the EU (or whatever favoured form of union) as a necessary conclusion. Even when we read the word "Europe" in an older context, we should remain prudent: altough Aristotle already used it as a geographical notion, it had little political and cultural content and remained desperately vague: the Asia Minor Greeks could be counted in or out depending on the context, and in the first century BC Sallustius could write of the modern Maghreb as far as Libya as being part of Europe. Also, note that I don't have my library at hand (damn me!): I won't quote anything I can't find the reference back, so I'll have to remain vague on several details. Above all, bear in mind that given the very large range of your question, any answer including mine will border on the endeavour of erudition more than on the kind of precise exercise that is the ideal on this sub.

That being said, onward to the real question: what were the main projects and visions that indeed proposed an unification of Europe as such? If I may, I will also restrict myself more to political projects, since Europe as a cultural space if both a much older (dating at least as far back as the VIIIth century, according to Pirenne's old these) and a much more porous reality.

The Renaissance and the invention of Europe

You mentioned Podebrady's project of connexio, pax, fraternita et concordia between christian nations; and luckily for us, it happens to be a great starting point. Before it, visions for an united continent had existed, indeed been partly realized in two institutions, the catholic Church and the Holy Roman Empire. However, as Peter Heather points out in his Restauration of Rome, both these institutions proposed themselves as exactly that: successors to the Roman Empire - that is, an empire both christian and universal. Yet, by the XVth century, both these claims had taken on serious water: far from being universal, the HRE was no longer even a dominant power in itself on the continent; and with the Great Western Schism and the Hussite wars, and soon Luther's Reformation, religious observance could no longer be what defined a common identity. Mark Greengrass speak of this era as that of Christendom destroyed: the old definitions of a common identity collapsed, and new ones had to be invented. As John Hale writes in the civilisation of europe in the Renaissance, a work you can refer yourself to along with Greengrass' for reference on the matters of European culture and identities at the time, it was in the XVIth century that geographers began using "Europe" instead of "Christendom" to designate the part of the world we are used to calling so. The Renaissance, in short, was the time people all over the continent had to rethink what their international and cultural community meant, and came up with "Europe" as an answer. A spectacular incarnation of this change of representations is to be found in the realm of cartography: whereas in the Middle Ages maps with symbolic meanings centered on Jerusalem (a christian, universalist depiction of the world), by 1570 the Europa Regina could focus on entirely on Europe as such and display its supposed superiority.

Now, when will he finally get back to Podebrady, you ask? Well, Podebrady was a hussite king, viewed as a heretic by the Church of Rome, and in poor terms with the Emperor. As such, his project for an union between european countries was the first that neither mentioned the Church nor the Emperor as keystones of unity. As such, it can rightfully be seen as groundbreaking in this regard. Yet, it also retained a sense of christian unity: the king had proposed it to put himself forward as a real christian (as opposed to a heretic) in the first place, and as such it devoted extensive attention to projects of crusade against the Turks. At the same time, the project neither encompassed every european country (Iberia, in particular, was supposed to join at a later, unspecified point), nor limited itself to it, since it left the door open to adhesion by muslim countries. Underlying this latter point was the fact that it was before all a "pacifist" project, not one ultimately based on a perceived common identity like a modern nation-state. As such, the bohemian king's project somewhat beautifully stands out as both out-of his time, or timeless - as it didn't stand a chance of realising its peaceful ideals - and grounded in the debates and interrogations of its time. I hate myself for not having my copy of Boucheron's Histoire du monde au XVe siècle at hand, but if you can get your hand on it and if you read french (comme ce doit certainement être le cas), rush for it and for its pages on the topic.

As the Renaissance drew to a close, contradicting movements expressed themselves. On the one hand, Europe had acquired a new sense of cultural unity through the humanist movement, as forever embodied by Erasmus, who traveled his life long across the continent and made a "purified" latin the international language of a culture nourished by ancient models; and it came seemingly close to political unification under emperor Charles V, who used in his propaganda the vision of a restauration of an universal empire encompassing all of the continent. On the other hand, Europeans increasingly questioned their own values as violence swept across the continent. The discovery and conquest of the New World both gave Europe a mirror of itself and its power and, as Europeans comitted atrocities and encountered new systems of values, led to increased questionings of european "civilisation". Montaigne, for instance, was both a brillantissime example of the highly-learned, travelling european humanist partaking to the common classical culture, and a cultural relativist which openly asked how Europeans could call themselves superior to Amerindians. Andrés Laguna de Segovia, as well, both called for peace and union across the continent in Europe the Self-Tormentor and questionned the continent's values in his relation of travels to Turkey.

Politically, the calls for continental peace had become a locus communis over the course of the XVIth century: for instance, the 1518 treaty of London had proclaimed a never-ending peace between every european ruler. Yet that peace was broken less than a year later, and the dreadful wars of religion engulfed Europe, culminating in the Thirty Years War from 1618 to 1648.