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Does a gentile who wishes to convert to Judaism have to follow the Mosaic law (addressed to Israel) or only the Noachide law (addressed to all humanity). I ask this question due to the apparent contradiction between dietary laws.
Also, does the Tanakh ever say that gentiles can enter the world to come, or judged for their sins?
Gentiles who wish to convert to Judaism follow Noachide laws, and as they learn more gradually follow more of Mosaic laws (called halacha). Until they have formally converted, they are forbidden to fully observe the shabbat so typically transgress one law on Shabbat to respect this.
I am not clear which contradiction in dietary laws you are referring to and it might be worth editing your question to clarify.
The Rambam (one of the greatest codifiers of Jewish law) writes explicitely (MT Melachim 8:11) that
Anyone who accepts upon himself and carefully observes the Seven Commandments is of the Righteous of the Nations of the World and has a portion in the World to Come. This is as long as he accepts and performs them because (he truly believes that) it was the Holy One, Blessed Be He, Who commanded them in the Torah, and that is was through Moses our Teacher we were informed that the Sons of Noah had already been commanded to observe them.
Anyone interested in converting should approach a rabbi and discuss the proper path to do so.
The answer to your question depends on what you mean by "convert" to Judaism. In general, when we talk about conversion to Judaism we are referring to the process of becoming a member of the Children of Israel. Such a person is obligated to follow all of the Mosaic laws as he is considered the same as any other Jew.
However, I have personally long distinguished between the Jewish religion and the Jewish nation. What we consider "conversion" nowadays involves joining the Jewish nation, which requires the observance of all 613 commandments. But one could convert to accepting the Jewish system of beliefs without actually joining the Jewish nation.
For example, suppose a practicing Christian realized that he really doesn't believe in the Christian understanding of God, but actually the Jewish understanding seems right. He does some studying and realizes the truth of the Torah and the Jewish religion. There is no requirement that this person join the Jewish nation and follow all 613 commandments. Instead, he could choose to remain a gentile and follow the seven Noachide laws, and he would be fulfilling everything that is expected of him. According to Rambam, such a person would merit entering the world to come. (See Mishneh Torah, Sefer Melachim uMilchamot 8:11.
