Hi,

I believe that ostensibly the Norse members of the Guard did convert, as most of Scandinavia was being forcibly converted by the likes of Olaf Trygvason and Olaf the Saint around this time quite a few may have already adopted Christianity as their main religion. Having said that, I think (and this is only from my research) that the Byzantines were relatively accommodating in turning a blind eye to some aspects of the Scandinavian's behaviours.

The Varangian Guard were formed c.988 AD by Basil II and were initially formed of men from Kievan Rus who had recently converted to Christianity in agreement between Vladimir I of Kiev and Basil. It is likely that most Guardsmen joining were baptised but, as with a lot of Scandinavians, they may have continued to worship older Gods from the Norse pantheon.

Examples of crucifix amulets that also resemble Thor's hammer have been found and highlight the ambiguity in many ex-Pagans religion around this time.

As for close friends to the Emperor, look no further than Harald Hardrada. He was the Domestikos of the Varangian Guard after giving several years of service to Byzantium whilst in exile from Norway. It's a very interesting story and he was not always in the Emperor's favour!

https://en.wikipedia.org/wiki/Harald_Hardrada

In fact, his story is very interesting from birth right through to his death in 1066 when he made his play for the English throne.

When the Rus were members of the formal guard they definitely formally converted. The Rus had formally installed themselves in Kiev and agreed to convert formally before the creation of the guard. (There is a story that Vladmir considered converting to Islam but didn't want to give up his vodka.) I'm sure individual members kept their paganism for awhile. But we know how powerful the Orthodox religion and customs were to the Rus. As they went onto to install it in Kiev and later Moscow.

However, they served the emperor long before that (for almost a hundred years) and there was no formal conversion requirement in place to do so. Really, the emperor needed them too much to try to force that on them.

Of course, later the more Western Nordic peoples joined the guard including men like, Harald Hardrada. But this is around the same time conversion is happening in the West. So, slowly, the old ways are already dying out. As far as I know it wasn't a requirement to formally convert but by then most of these people would be nominally Christian in some form anyway.

In the early days, (the late 800s into the 900s) The Norse were often fairly impressed by Orthodox Christianity in a way they weren't originally by Western Christianity. For a period of time Western Christianity was sort of the religion of losers from their POV. (Seeing as how they kept losing to them fairly easily for awhile.) But Orthodox Christianity was large and imposing. They had beautiful churches like Hagia Sophia (and there is another story of a Viking "queen" who saw it and converted on the spot because God must dwell there). Although the Byzantines suffered their losses and were contemptible in some ways they were generally considered a successful, imposing people who commanded a lot of respect and awe from the Nordic people. So, conversion wasn't quite as arduous as it was in the West.