Short answer is, we don't really know the full extent of it. Some Byzantine religious-cultural influence existed but organized orthodox institutions were not established.
Scandinavians started travelling out in to the world to a far greater extent as the Viking Age go started in the late 8th century; and no less so with the 'eastern route' (Austrvegr and similar expressions) or rather routes-plural from the Baltic on the rivers through the Rus' kingdoms to Byzantium. Even if the journeys on the 'western route' to England and France are more well known today. This isn't a local phenomenon to Scandinavia either, the founding and growth of Scandinavian trading posts like Birka (outside Stockholm) and Hedeby (at the foot of the Jutland peninsula) were mirrored by the founding and/or growth of towns along the route such as Staraya Ladoga, Veliky Novgorod and Kiev. Scandinavians were going off to serve the Byzantine emperor well into the 12th century, so in this respect one might consider the Viking Age to have gone on longer in the east than in the west, where it's usually bookended by the Norwegian defeat at Stamford Bridge in 1066.
As you're aware, these contacts are well known and there is a great wealth of evidence for these contacts; coins and artefacts, rune stones memorializing men who died in the aforementioned places (and even specific, deadly parts of the Dniepr Rapids) and also the post-Viking Age written accounts. They all show that the east-travellers predominantly hailed from the easternmost parts of Scandinavia: More or less what's now Stockholm County, and the island of Gotland. Which is pretty well expected, geographically, and also historically considering that Stockholm and Visby on Gotland would arise as major Hanseatic trading towns in the 12 and 1300s.
However, there is little evidence to go on when it comes to religious influence. The problem here is that the main accounts that we have of missionary activity in Scandinavia are from a single source; the dioscese of Hamburg-Bremen, in the form of Rimbert's Chronicle and the vita of Saint Ansgar (9th century) and later the ecclesiastical history written by Adam of Bremen (11th century). Particularly the latter source is not expected to have given due credit to the English and Irish missionaries which were quite active in Sweden in the 11th and 12th centuries. E.g. martyred missionaries such as saints Botvid, Eskil, David of Munktorp and Sigfrid were all supposedly English. Sigfrid had supposedly even been the archbishop of York under the femininely-named king Mildred of England before leaving for Scandinavia. (although English sources mention no figures by those names) But even Adam does at least mention English missions, and the saints left cults behind.
There's no record of orthodox missions though, which does not entirely exclude their existence (even if unlikely). Finland's first known missionary is Saint Henry in the 12 - another martyred Englishman. Yet there is evidence that Christianity first reached Finnic peoples from Novgorod in the east, in that a number of basic words pertaining to Christianity, like the words for "bible" and "crucifix" are of Greek-via-Slavic origins and not Latin-via-Germanic. On the other hand, some Finnish peoples (e.g. Karelians) were already under Novogordian rule and others were far closer than Sweden.
When it comes to archaeological evidence, there are as said many eastern objects have been found in for instance Birka and Sigtuna and other places around Stockholm, and that includes Byzantine and Rus' crucifixes and other artefacts - but it's hard to interpret this as an expression of religious preference; many Islamic and even Buddhist objects have been found as well, without necessarily having been valued as anything other than as objets d'art. Eastern influence in Scandinavia also peaks in the 970s, which is before east Sweden had converted. When conversion comes to the Stockholm region in the 11th century, we have a popular trend of raising rune stones as Christian monuments, resulting in Sweden having nearly 3,000 Viking Age stones, versus - in round figures - about 250 in Denmark, 150 in Norway and 80 in the rest of the world. It is these Christian 11th century stones that make up most of the stark difference here. So one of the ways rune stones may give a clue here is that they're a popular expression of people's Christian faith. As opposed to for instance the Jelling Stones in Denmark, a one-off commissioned by the king to announce the christening of the entire kingdom, we now have stones that seem to advertise the faith of individual families, and some more middle-class ones, not just royalty. And one interpretation of the trend is just that: That Jelling started a trend which spread geographically and 'trickled down' class-wise to become a popular thing in Sweden a century later.
A different interpretation was given by Linn Lager, who instead argued the stones represented a kind of protest against Hamburg-Bremen's bishops and missionaries and a nationalistic demonstration in favor of the Christianity they'd been 'sold on' by the English missionaries. The linchpin of the argument being that the crosses on rune stones are, in Linn's view, inspried by Anglo-Saxon stone crosses, and that Denmark - which had more German missionaries - has fewer stones. A problem with this, though, is that Norway has even fewer stones while having more Anglo-Saxon missionary presence than Sweden did. There are also other theories such as the stones serving to document inheritance but the point is that there are no eastern influences on this early domestic Christian art that comes abut as Christianity is reaching the broader populace in east Sweden.
Gotland is different though.
In many respects Gotland is and was culturally quite distinct. Their language (Old Gutnish) was its own branch of the North Germanic languages, separate from Old Norse although probably mutually intelligible with it. They had local myths, recorded in the Gutasaga, not attested elsewhere. They had their own tradition carving picture stones from the limestone the island is made out of (while runestone carving arrived there relatively late), and so forth.
On Gotland, prominent Byzantine influence on the religious art remained in the 1100s and even into the 1200s. For instance, this angel in a mural from Källunge church, mid-1100s, or this baptismal font from Gardar (late 1100s) by the master that later historians named 'Byzantios' because his work was Byzantine-influenced (although not necessarily first-hand influence). This is together with Romanesque styles and older Norse influences, and the consensus on these beingm Byzantine influences are pretty broad.
There's less support for Byzantine influences on the mainland, although some have argued for instance for such influences on the murals on Forshem church in Västergötland, and the architecture of Saint Olof's church ruin in Sigtuna. Some years ago a paper argued that so-called [lily stones (like this) from 11th century Västergötland were also of Byzantine influence, although that paper was criticized pretty roundly.
To summarize, the contacts were there and the cultural influence was felt strongly at least in Gotland. But we do not have much in the way of written sources from present day Sweden until the 13th century (and not that much in that century either). We have more Icelandic and Norwegian texts, but these - as one might expect- do not reveal much eastern religious (and cultural) influence but more from England and France. The oldest book in Sweden, the Skara Missal, was probably written in north France in the mid 1100s. Not to go through the whole body of literature - the short of it is that there's no eastern influence on any of the preserved liturgy.
There's no doubt that out of the people from Gotland and Uppland and elsewhere who went off and - for instance - served in the Varangian Guard down in Constantinople, many came back with the religion of their former employers. The Viking Age crucifixes in Byzantine style were probably not worn just as fashion accessories. But if this left any mark on Christianity by the time the general population converted is harder to say. We do know these conversions abroad left a mark on the influence of the pre-Christian religion. For instance, not long after the Viking Age started and people began coming home with crucifix necklaces, people began to wear ornate Thor's hammer pendants in a similar way.
To some extent it may also be a moot point. These were converts, not clerics or theologians. There's not much reason to think most received much in-depth instruction. They returned to a country without churches (some would build their own family chapels with the money they gained abroad), without priests, without bibles or anyone who could read one anyway, and they no doubt engaged in many unorthodox and syncretistic practices that'd later be deemed unchristian.
So from that perspective, neither a 'orthodox' or 'catholic' label would be apt as it implies adherence to a more detailed doctrine and organized form of the religion than existed then and there.
