Slavic is a language family and to qualify for a slav (outside of 19th century essentialist slavophile talk) you only need to speak a language which belongs to the slavic language family.

What was it with the East, West and South Slavic language sub-families?

Well!

Think of the word "garden", with the "gord" root ("town" in the slavic languages): in the west and south slavic language families out of that word you got "grod" or "grad". In the East slavic family you got "gorod". And the same happened in every "vowel + L or R" combination. Here East is different from the rest. (Czech: mleko, Serbian: mleko, Ukrainian: moloko, English: milk)

The second major difference are sounds which were originally "kw" or "gw": in east and south slavic languages they turned into "tsv" or "zv". Here the West slavic languages are different from the rest: they retained the two sounds mostly unchanged. (Russian: Zvezda, Bulgarian: Zvezda, Polish: Gwiazda, English: Star)

Source: Roland Sussex and Paul Cubberley - the Slavic languages (Cambridge university press).

If Rurik already did speak a Slavic language we've no idea what it was like. However, two centuries after his death we get the first inscriptions from the city from which he ruled Russia, from Novgorod. And it's in the Old Novgorodian dialect. From the above it has both the features which set the East Slavic languages apart and those which set the West Slavic languages apart, which lead some of the scholars to assign it to a special "North slavic" branch. So: if prince Rurik was some kind of a slav he was nothing like the slavs we know from later.

A "Slav" is simply a member of an ethnic group defined by historically speaking a Slavic language such as Polish, Russian, or Serbian. Strictly speaking, then, this is technically more of a question of linguistics; in brief, Slavic languages share similar syntax, phonology, and other linguistic features, and are most likely all ultimately descended from a much older language retroactively called Proto-Slavic.

Rurik was a figure of debatable historicity (referenced primarily in a 12th-century chronicle) who established a state in eastern Europe, known as the Kievan Rus'. Supposedly, he was invited by local warring tribes to restore order, and his legend was built up by his successors, who claimed descent from him as evidence of their legitimacy as rulers. The tribes I mentioned, however, are the real meat of your question--these tribes were Slavic tribes, speaking archaic versions of the Slavic languages. Rurik was never a Slav, except perhaps honorarily, and he certainly was not an ethnic Slav; as time passed, the rulers of the Kievan Rus' adopted the Slavic customs of those they ruled and became more or less indistinguishable from the locals.

It's not entirely clear at what point Slavs came to inhabit modern-day Russia; we have few written sources about the area until the medieval period, save Roman sources (and Roman ideas of linguistics and naming conventions often make it hard to ascertain who they were talking about.) Archaeological finds suggest a Slavic homeland in western Ukraine. During the Migration period--roughly corresponding to the fall of the Western Roman Empire through the start of the Early Middle Ages--Germanic tribes moved out of Eastern Europe into Roman territory; Slavs seem to have subsequently migrated west into the now-Germanic-tribe-free areas of Eastern Europe, like Poland. During this period and throughout much of the early Middle Ages--perhaps bleeding into the High Middle Ages, depending on how you classify it--they also gradually seem to have migrated north into Russia 'proper', displacing local Finno-Ugric tribes in the process.

Edit: A word

As for theories, the very existance of rurik is a questionable thing.

His calling in the primary chronicle is a legend. Widukind of Corvey has the Britons call the Saxons using the very same language ("our land is rich and spacious, but miseri Bretti crebris hostium incursionibus fatigati et admodum contriti.") and the Korean chronicles contain things of that sort as well.

It's a legend which legitimized the ruling dynasty.

Now, you could say the Ryurikid dynasty did exist and therefore there has been someone named Ryurik. The problem with that would be that it didn't call itself Ryurikid until the 15th century.

As for it's ethnic composition, the Russian leaders of the historical era had mostly come from other ruling families. Andrey Bogolyubski (12th century) has, for instance, been half Polovitsian (related to khan Aepa) and quarter-English (related to king Harold Godwinson) and, to a one-eigth part, a Greek (related to the Byzantine Emperor Constantine IX.)

You could well say they have all, ethnically, been nobles with a stiff upper lip. And they have been polyglots as well. Vsevolod I. of Kiev (11th century) is recorded to have spoken 5 languages.

Now, Ryurik isn't described to have created the Kievan Rus. The chronicle has him reside in Novgorod. He is said to have been called "from beyond the sea" and he is identified as a "Varangian" ("these Varangians were called Rus, just as other ones are called Normans and Angles and other ones Goths"). His son Oleg is said to be the one to have conquered Kiev and to have moved the capital there.

Source: An interview with dr. N. I. Danilevski - Ryurik is a legend.

Let me ask a follow-up: based on contemporary knowledge of haplogroups and genetics, is it fair to say that members of ethnic groups that are geographically close to Slavs but who are not themselves considered Slavs (such as Tatars, Hungarians, Romanians/Moldovans, Albanians, Greeks, Lithuanians and Latvians) are, in fact, genetically distinct from Slavs? Or are these adjacent peoples essentially genetically indistinguishable from their neighbors, with the Slav/non-Slav distinction being purely cultural and linguistic?