Well, first, they don't really need to be distinct; if they regarded themselves as different peoples then they were different peoples no matter how similar someone else thinks they are.

"Swedes" is too present day, however, as all contemporary sources distinguish Svear (Swedes-proper, Svíar) and Götar (Gautar/Old English Geats) and often also Gotlanders (Gutar). Other than the last, those main nations are further divided into more local groups such as Västergötland/Östergötland (Götaland), Uppland and Södermanland (Svealand), Jutlanders and Scanians (Denmark), Rogaland and Viken (Norway). These areas had different laws and customs (the latter refleced in the former of course), even if the first provincial laws, like so much else, were written down a few centuries after the Viking Age.

Many differences that exist would not likely be reflected in the Saga literature, because it largely concerns itself with the nobles and elite class, who intermarried, raided and traveled together, but also formed alliances and fought wars against each other. Scandinavian societies were dominated by these groups and strife between them, between nobles and kings who wanted to centralize power, post-Christianity you also have the bishops and the church as an additional power, and all this continued for centuries into the Middle Ages; or even throughout the Middle Ages. So in the Viking Age a war between a king of Norway and a king of Denmark is best described as precisely that, and not as war between Norway and Denmark.

So what you're asking here are very broad but specific questions about an era that's prehistoric for the region. There are no written sources from Viking Age Scandinavia, while the sources written down later, mainly in Iceland, are mainly concerned with a pan-Scandinavian elite, who don't just appear rather homogeneous in the sagas but to an extent also in the archaeological evidence. Fancy ship burials are rather similar from Norway to Sweden, while ordinary people's burials vary a great deal within the countries - stone mound, earth mound, no mound, cremation or no cremation, position of the bodies etc. (here's a poster, Anders Andrén has studied in more detail the burial customs within Sweden)

From an outside view, there was not necessarily much reason to make a distinction since as said, they raided together. The Great Heathen Army supposedly consisted of people from all parts of Scandinavia. Some of the conflation may have come later though; as e.g. the term "danegeld" was not used in the 11th century, Svealand runestones speak of exacting tribute in England, while most Anglo-Saxon coins have been found in Gotland. There's also the references in English sources to 'Danish axes', even though the Scandinavian axe types are really that distinctive; Bayeux similar shaped axes existed in England (tons on the Bayeux tapestry) but also Russia and the Baltics too.

Were the Norwegians more geared towards the Vanir? Why?

It's debated how significant the Æsir-Vanir distinction was, to the extent that it's even questioned whether it actually existed in the first place, or whether 'vanír' was just a less common word for gods. But, contrary as it is often portrayed, the religion was by no means homogenous nor static in time. For instance, there are no cult-related place names for Týr outside of Denmark, and even more complicated, the name Týr often appears as a generic god name. On the other hand there are no place-names for Ullr in Denmark, despite being very frequent in Sweden and Norway. Both Ullr and Freyr are very underrepresented in the Eddas compared to how many cult sites there are, while popular Eddic figures like Loki and Heimdallr have no place names, and no evidence of Loki being around prior to about the 10th century. (For more details and numbers, see e.g. Stefan Brink's paper How uniform was the Old Norse religion? )

Were Swedes overrepresented in the Varangian Guard?

Swedes certainly traveled east (by which we really mean the southeast, down the Dnepr to the Black Sea) to a larger extent while Danes and Norwegians traveled west more to England, France, Ireland etc. The Rus' also possibly/probably originated there, and trade and political contacts with Novgorod and Uppland existed in the 11th century. There are runestones in Sweden bearing witness of those who've gone to Greece and Byzantium, and most importantly a lot of Byzantine and Arabic finds there. Some marks were famously left down there as well, such as the runes in the Haga Sofia but more importantly the Pireaus Lion which was defaced by Swedes from Roslagen.

Were Norwegians more likely to settle elsewhere,

I don't think one could say that when there's really only one data point for that namely Iceland/Greenland. In that case the settlers are thought to have originated mainly from northern Norway and in any case certainly from the west coast. So even that one case isn't necessarily representative of all Norway.

but Danes more likely to be politically involved?

I don't know what that means.

Regarding weapons, swords were exclusive weapons because they were exclusively a weapon. They were only for killing people with, not usable for hunting or anything else. Hence, affording a sword was a sign of status.

Did Norwegian warriors use axes more commonly?

There's no reason to think that. I don't know that there really are more axes found in norway, or that that'd be representative of more axe ownership. Axes were never used solely for fighting with; the type of axes thought to perhaps have been a prestigous/warrior axe, the Petersen Type M remained in use in Norway as a 'peasant axe' until the early modern period (example).

Is it related to an old timber industry/a logger lifestyle?

There was no timber industry or loggers in the Viking Age Sweden or Norway. If they needed wood they'd go chop down a tree. Logging beyond what was needed for self-sufficiency would have to wait until there was a Baltic trade network to sell it. In any case, Sweden and Finland had a larger logging industry and the traditional building practices are not very distinct other than in the fine details.

Why does Sweden have so many runestones, nearly ten times more than both Denmark and Norway combined? Why aren't there many (or any?) runestones in places like pre-Christian Saxony and Frisia?

The answer to the former might answer part of your later question, since in sheer numbers, most runestones are christian monuments. The majority of the ones in Sweden, and ones in total, are mid-11th century, Christian, Younger Futhark inscriptions with Urnes style decorations. Typically raised in memory of relatives and husbands who've died. Both ones who've died at home and died abroad on Viking journeys. The style is slightly different but this is also true of most runestones on Bornholm in Denmark, the runestone-densest region there.

Götaland was already at that time using other monuments such as Eskilstuna-coffins (frequently with runic inscriptions) and Lily stones (without any inscriptions) and other grave ornaments. Eventually continental-style stone slabs would win out as high-status grave markers. runic inscriptions continued to be made for centuries past the Viking Age though. We do not know why there was a sudden rise in popularity of this type in the mid-11th century Svealand (and Uppland in particular) but it coincides with the christening of the area, and may have been a way to announce and advertise the introduction of the new religion. (the Frösö stone, far up in Jämtland is explicitly that) The powerful family of Jarlabanke in Täby north of Stockholm alone erected about 20 stones, so many that much of his family tree could be reconstructed from them.

Pre-Viking Age runestones may be far less common but they are also more evenly distributed, and in some areas (e.g. Värmland) there are only such stones.

Why does Norway have so many stave churches compared to the rest of Scandinavia?

We don't know that Norway does. With 28 of them, Norway has more standing stave churches than the rest of Scandinavia. But that is still only a couple percent of the original number. Stave churches are known from Gotland, Västergötland in Sweden and Skåne in (then) Denmark as well. So it would not take a huge difference in the conservation rate for Norway to have as many as it does. On the other hand, Norway no Round Churches unlike Denmark and Sweden, which are 12th century stone churches which doubled as defensive structures. Perhaps Norway was safer.