I mentioned these tribes as passing examples. If you want to research how exactly Islam spread in arabia at the time of the Prophet pbuh, it is best to start on page 1 rather than throwing onself into the middle of the book and trying to piece together the story. I'm more than happy to provide you with more information about these groups.
As I said earlier it was not a matter of the Muslims suddenly cropping up in the middle of arabia and everyone rushing to attack them. The Muslims were intially driven out of their homes in Makkah after being persecuted, so their conflict was initially between them and the Quraysh, while other tribes in the region watched to see how events would play themselves out. Rather than creating more conflicts, the Prophet pbuh sent delegations to many other groups for the purpose of peace treaties and creating allies. When the other arab tribes saw that a band of vagabonds, chased out of their homes and persecuted, had soon after triumphed over one of their oppressors, one of the most prominent tribes in Arabia, many more of them allied with the Muslims and opened themselves up to his teachings. The peaceful alliances were the cause of many more to enter into Islam and accept the simple and basic Islamic creed. They had witnessed the miraculous triumph of the Prophet Muhammad and they saw that all he called to
was in-tune with the natural human disposition. There are over sixty tribes that sent delegations to the Prophet pbuh after his victory in Makkah.
About Najrân, Ibn Ishaq in his Sirah (biography of the Prophet) stated:
'When the delegation of Najrani Christians came to the Prophet at Madinah, they entered his mosque in the afternoon to meet him. It was their prayer time, so they began to perform their prayer in the mosque. Some Muslims were about to prevent them from doing so, but the Prophet, upon whom be peace, said, "Let them pray.", So they faced eastward and performed their prayer.'
Showing again, the unique tolerance of the Prophet Muhammad pbuh.
As for what you quoted on Taghlub, this is clearly speaking about the time way before the Prophet pbuh came along. Ibn Kathir is providing in his book the explanation of the historians of how the Jewish and Christian tribes came to settle in arabia and he is correlating it with the Qur'anic verses. By the time the Prophet pbuh came along, there were Jewish tribes settled in Madinah, and a few christian tribes, like Banu Taghlub in northern arabia.
Coming to the Jewish tribes in Madinah, the Aws and the Khazraj has always been the political authorities in Madinah, and once they accepted Islam, their tribal rivalry and fighting ceased and the head political authority was now the Muslims. Dr. Hamidullah comments on the treaty made:
When the Prophet Mohammed settled down in Medina, he found there complete anarchy, the region having never known before either a State or a king to unite the tribes torn by internecine feuds. In just a few weeks, he succeeded in rallying all the inhabitants of the region into order. He constituted a city state, in which Muslims, Jews, pagan Arabs and also probably a small number of Christians, all entered into a statal organism by means of a social contract. The constitutional law of this first 'Muslim' State - which was the confederacy as a sequence of the multiplicity of the population groups - has come down to us in toto, and we read therein not only in clause 25: "to Muslims their religion, and to Jews their religion," or, "that there would be benevolence and justice," but even the unexpected passage in the same clause 25: "the Jews . . . are a community (in alliance) with - according Ibn Hisham and in the version of Abu-'Ubaid, a community (forming part) of - the believers (i.e., Muslims)." The very fact that, at the time of the constitution of this city-state, the autonomous Jewish villages acceded of their free will to the confederal State, and recognized Muhammad as their supreme political head, implies in our opinion that the non-Muslim subjects possessed the right of votes in the election of the head of the Muslim State, at least in so far as the political life of the country was concerned. (Hamidullah, Introduction to Islam, paragraphs 414-416)