There is a perception that Judiasm has no interest in seeking converts, but actually, ancient Jews actively sought converts, believing that G-d sent Jews into exile to bring people to Judiasm. Despite the fact that conversion was illegal in Rome after it became Christian, and also in Muslim communities, it persisted. However, by the late middle ages the Jewish community was discouraging converts because the danger was too great.* One thing we are encouraged as Jews is to "multiply like the stars in the sky" -- I think this idea of building the ranks can extend to a welcoming of converts.
In present day, there is debate whether or not people should be allowed to convert unless they are willing to practice as fully observant Orthodox Jews -- but most conservative and reform rabbis require less to varying degrees -- largely a sincere committment to living as a practicing Jew, raising a Jewish family, circumsicion for the male (Which most in the US are anyway), some level of education in the history, laws and language, and a ritual bath.
Abraham entered G-d's covenant at 99, and there is also a saying that converts are dearer than the Jews who stood before Mt Sinai - because they did not require "proof" -- or because they actively choose to be part of the faith, as opposed to being born into it. *
Growing up, we were taught that converts are as Jewish as the rest of us born into Judiasm, if not more so. Now, whether or not converts are treated as such is another story. And while "goy" is not a slur in itself, it can be used as such. As any word can, I suppose. It does carry the stigma of being an "outsider" in any case (IMO), but it may vary from country to country?
*most of this paraphrased from the book Jewish Literacy by Rabbi Telushkin
I am not sure that converts are turned away 3x -- but I would assume it is to ensure that they are sincere and committed to the conversion and not acting on whim. Isn't there also a tradition of apologizing 3x for an act? Or offering a gift / food / etc. to a guest 3x before it is actually considered "refused" in Eastern cultures?
In any case, apologies for the long response. Hope it helps!