Did he really exist?

The answer, as you might suspect, is that it is unclear.

Our earliest surviving reference to Rurik comes from the Russian Primary Chronicle which was compiled by Russian monks between 1037 AD and 1118 AD.

However, according to the chronicle, Rurik enters history when he devastates the settlement of Staraya Ladoga in the year 862. From there, he goes on to settle in Novgorod and lead that city until his death in 879.

So, the Primary Chronicle is recounting a story that is at least 150 years old when it is written down. Also, there are numerous areas in the early sections that seem to have been presented in order to legitimize the rule of the Rurikid dynasty. The most notable example would be the narrative of the "invitation of the Varangians" which as you suspect, is a way to legitimize the contemporary Grand Princes of Kievan Rus.

On the other hand, the Byzantine chronicle De Administrando Imperio written in the mid-10th century refers to Igor, the supposed son of Rurik, as well as mentioning Sviatoslav, the grandson of Rurik, who was alive at the time the chronicle was compiled.

how did he establish his rule over the Slavs?

So, there is not very much doubt that there was Norse settlement along the river Dnepr and the establishment of trading settlements at Staraya Ladoga, Novgorod, Kiev and elsewhere.

However, there are two schools of thought about how important this Norse element was, and what exact role they had in the development of the Kievan Rus state.

The Normanist school holds to a literal reading of the Primary Chronicle, and insists that cohesive groups of Norse traders and adventurers established themselves as ruling Princes at these trading centers, ruling over Slavic people unable to rule themselves.

The anti-Normanist school holds that Norsemen arrived in small waves of immigration, and were not large enough to establish themselves as rulers over the native slave population. This school posits that the Norsemen primarily engaged as traders, and quickly married in to local Slavic notable families, and Norse influence was quickly assimilated into the local Slavic culture.

An important consideration in weighing these two competing schools is the recognition that in the 8th and 9th centuries, the Khazar khaganate had extensive power along the lower reaches of the Don and Dnepr rivers. This muslim Khaganate had extensive trade contacts with the Abbassid caliphate, and the trade in Abbassid silver seems to have been an early driver of Norse exploration of Rus lands. An established power in the lower reaches of the Dnepr would seem to leave little room for the establishment of new governments by encroaching Scandinavian armies. And yet, by the mid 10th century Kievan Rus was at least influential enough not to acknowledge Khazar overlordship.

So, the Russianist historian Wladyslaw Duckzo posits that Slavic peoples observed Khazar forms of social organization, and established a Rus Khaganate by the 830s, prior to the date that the Primary Chronicle gives for Rurik's arrival at Ladoga by some 20 years.

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1. Viking Rus: studies on the presence of Scandinavians in Eastern Europe By Wladislaw Duckzo. Chapter 2 explores his hypothesized Rus Khaganate.

2)Medieval Lands by Charles Cawley. This is an uploaded version of his chapter about Rurik and the Rurikid dynasty. In it, he addresses some criticisms of the Primary Chronicle.