Here is an excerpt from Russian Primary Chronicle (I am not sure why it's traditionally translated this way; the original title is: Povest' Vremennykh Lyet, which means: Story of Years Gone By):
6368–6370 (860–862). The tributaries of the Varangians [Vikings] drove them back beyond the sea and, refusing them further tribute, set out to govern themselves. There was no law among them, but tribe rose against tribe. Discord thus ensued among them, and they began to war one against he another. They said to themselves, “Let us seek a prince who may rule over us and judge us according to the Law.” They accordingly went overseas to the Varangian Rus': these particular Varangians were known as Rus', just as some are called Swedes, and others Normans, Angles, and Gotlanders, for they were thus named. The Chuds, the Slavs, the Krivichians, and the Ves' then said to the people of Rus', “Our land is great and rich, but there is no order in it. Come to rule and reign over us.” They thus selected three brothers, with their kinsfolk, who took with them all the Rus' and migrated. The oldest, Rurik, located himself in Novgorod; the second, Sineus, at Beloozero; and the third, Truvor, in Izborsk. On account of these Varangians, the district of Novgorod became known as the land of Rus'. The present inhabitants of Novgorod are descended from the Varangian race, but aforetime they were Slavs.
You are right in being suspicious.
The first version of the chronicle was compiled by a monk named Nestor the Chronicler, employed by Sviatopolk (Michael) II of Kiev, well known for his pro-Scandinavian views and policies. Sviatopolk was the grandson of the famed Varangian ruler Yaroslav the Wise, one of the best known rulers of pure Varangian stock. Sviatopolk's father (Iziaslav I) was himselfs half-Varangian and half-Swede, a full bloodied Viking.
Nestor presents Yaroslav as a model of virtue, despite what little we know of his life from other sources hint at the fact he was your typical Varangian ruler, being mired in endless feuds with his blood relatives and wars with any nationality under the sun (including a raid on Constantinople which ended in defeat).
It's well possible Nestor may have used the "mythical" (as opposed to real life) Yaroslav as a model for the previous rulers of Varangian stock, especially considering the fact the Primary Chronicle itself has very little to say about the first great ruler of Rus, the semi-mythical Sviatoslav (we know he really existed but his life is shrouded in mystery and legends), hinting at the fact the sources available to Nestor (lost to us except for the chronicles of John Malalas) were of dubious quality and he had to "fill the gaps" somehow.
Your point 2 may be addressed by looking at the life of Nestor's patron, Sviatopolk. Upon his death, the Slavonic inhabitants of Kiev rose in revolt against the Varangian officials and called upon the half-Byzantine Vladimir II Monomakh, who was delighted to take over the princedom but repaid (at least in part) the trust put into him by curbing the power of the Varangians in Kiev.
3. It's interesting that the "according to Law" is translated from "po pravu". That can be translated from Old Slavonic as "according to law", "according to custom", or "according to truth". It's not clear whether the Chronicle is saying that the Princes were invited to introduce some (presumably Swedish) custom into lawless Slavic lands, or that they were invited to figure out what the truth is (i.e., discover law naturally), or to create positive law de novo.
"Pravda" fundamentally means "that which is right", and "prav" means "right" or "correct". You have many derivates and derived meanings stemming from there one of which is "law" (but also justice, right, privilege...). So in a society which lived by customary law "po pravu" could refer to a custom, but only in the context of law. It can not mean "according to the custom of the Swedes" because to Old Slavs those customs aren't law. Only their own ones are. So to seek out a prince to judge "po pravu", would probably mean, seeking an arbiter to pass judgement on the existing customary law of the Slavs (there was no law among them when tribe rose against tribe, implying they had law it just wasn't being abided by and they needed a dissinterested, neutral foreign aribeter to restore it — that is at least according to the hagiographer trying to sell us the story), albeit you're right technically it could also mean seeking out somebody to administer justice based on "that which is right", ie to discover law naturally.
I am not sure why it's traditionally translated this way
While the most common name in Russian is indeed "Повесть временных лет", occassionally it is referred to as "Первоначальная летопись" - probably because it claims to describe the history of Russian land from the beginning. So I guess "primary" in this context means "first in order of time or development", not "of first rank, importance, or value".