Since I was magically summoned (!) by name I figured I'd throw some answers in. Strap in, this is my speciality and I'm writing a book on it, but I'll try to be gentle!

First off, it should be noted that the TV show Vikings has a tremendously loose chronology. Ragnar Lothbrok and his "brother" Rollo are both present at Lindisfarne in 793, and the siege of Paris which took place in 885, and this Rollo is ostensibly the same Rollo who founded Normandy in 911... so yeah, the time line is all kind of messed up (which is fine if what you want is entertainment rather than history, but that is a whole other post!).

That being said there are numerous contexts within which outside forces have an impact on Scandinavia during the Viking Age.

In terms of indirect impact and contact, trade is huge throughout the Viking age, this is a period where the first "urban" (or proto-urban as my archaeologist friends would demand I say) sites appear in Scandinavia. Places like Ribe and Hedeby (Haithabu/Schlewswig) in Denmark, Kaupang in Norway, and Birka in Sweden are all fundamentally centered on trade with the wider world, as evidenced by the number of foreign goods found there, including glassware, pottery, swords, jewelry, coins etc (check out D. Melleno "North Sea Networks" in Comitatus 45.1 2014 for more on this). It is also telling that some of the very first Viking raids abroad occur at key trade sites. It is very likely that these sites were already well known to Scandinavian traders who could just as easily become raiders if the pickings looked good. Dorestad, the key Frankish trade center, is sacked every year from 834 to 838, for instance.

Alongside trade comes ideas and technology. This is the period where we see the "invention" of coinage in Scandinavia (again at these same trade sites) and almost always this coinage looks remarkably like foreign coinage. The kings/chieftains involved in these sites are imitating what Frankish, Anglo-Saxon, and later German kings do and making a profit off it.

Another key idea that comes from outside, in part due to trade but also due to diplomacy is Christianity. The first Christian missionaries arrive from Francia (France/Germany) in the early/mid 9th century and though they aren't entirely successful it is clear that Christian ideas are making inroads well before the "official" conversion of the 10th/11th centuries. And throughout this whole time missionary activity from first the Carolingians and then the Ottonians continues chugging along in one form or another (you can read the various primary sources on this in translation, perhaps most easily in The Viking Age: A Reader from University of Toronto Press).

As for direct contact, there is evidence of diplomatic engagement between the Carolingians (Franks) and the Danes as early as the 780s. This is clearly the result of Carolingian expansion into Saxony which creates a shared border between the Franks and Danes by the turn of the 9th century.

Around this time (say 804) the Danes and Franks end up in a sort of weird proxy war over control/influence over the various Slavic tribes that site "between" them, and this continues on well into the Ottonian period.

By the mid 9th century the Franks are actively intervening in Danish politics, with pretty mixed success overall. As noted in the other answer I gave in Germany, linked by /u/quickspore the splitting up of the Carolingian empire fundamentally changes the calculus. Instead of one big scary empire on its borders the Danes (and other Scandinavians) now have fairly ripe picking in the form of a number of smaller Frankish kingdoms. Each of these kingdoms has different tactics and relationships with the Vikings but it is really only East Francia, the nucleus of the future Germany that has direct political contact. Meanwhile Vikings are spending time in Francia, sometimes as raiders, sometimes as traders, sometimes as settlers, and even sometimes as rulers of counties or sub-kingdoms (check out Coupland, "From poachers to gamekeepers" in Early Medieval Europe 7.1 1998 for a nice overview of the complex relationship).

Jumping forward, the Ottonians do in fact have very active engagement with the Danes. In fact Otto defeated the Danes and their slavic allies in the late 940s and invaded Denmark in 950 and maybe even forced Harald Bluetooth to convert (though the source on this, Adam of Bremen, is suspect, you can read him in a translation by Timothy Reuter).

So tl;dr: There is a tremendous amount of cross-cultural contact between Scandinavia and the continent. This contact takes place via trade, via diplomacy, via missionary activity, via osmosis, via diaspora and even via direct conflict in both directions. And our sense of these various connections keeps getting stronger the closer we look.

You may be interested in this post from a couple of years ago Did the Vikings ever pillage Germany.

/u/Mediaevumed discusses in brief sources and politics between Denmark and East Franconia. While /u/Kartoffelplotz discusses several raids, focusing on the “Great Heathen Army” of 880-882.