Are there non-Jews in Olam Haba (the world to come), either in Gehenom (Hell) or Gan Eden (Heaven)?
Yes. The Talmud (Sanhedrin 105a; Rosh Hashanah 17a; See also Tosefos on Sanhedrin 13b) states that the wicked people of all nations will go to Gehenom (Hell), and that righteous people of all nations, Jew and non-Jew alike, will got to Gan Eden (Heaven).
The Rambam (Maimonides) writes that anyone who has acquired knowledge of God and follows the Sheva Mitzvot B'nei Noach (7 Noahide Laws) is considered "righteous," and will go to Heaven. (Hilkhot M'lakhim 8:14; Hilkhot Teshuva 3:5)
Rav Yisrael Lipschutz, the Tiferet Yisrael, writes that even the average among the gentiles makes it to Heaven. (Yakhin, Sanhedrin 10:2) As he says,
Even without the holy words of our sages who told us this [i.e., that pious gentiles merit olam ha-ba], we would know this from our intellect because “God is just in all His ways and benevolent in all His deeds.” We see that many pious gentiles recognize the Creator, believe in the divinity of Scripture, act compassionately toward Israel, and some have done great things for entire world.
Could you imagine that these great deeds will not be rewarded in olam ha-ba? God does not withhold the reward of any creature. Even if you say that these pious ones who keep the seven Noachide commandments would not have the status of a ger toshav (resident alien) because they never made a formal acceptance before a court or because we do not accept gerei toshav in our day, since they do not act like Esau, they have a portion in olam ha-ba. (Translated by Rav Yitzchak Blau)
For an extended an in-depth treatment of this topic, please see this article by R' Gil Student.
Yes, gentiles can go to the world to come.
This states that they can as long as they're not wicked.
Talmud, Tractate Avodah Zara 10b
Once he[Antonius] asked him[Rabbi]: 'Shall I enter the world to come?' 'Yes!' said Rabbi. 'But,' said Antoninus, 'is it not written, There will be no remnant to the house of Esau?' 'That,' he[Rabbi] replied. 'applies only to those whose evil deeds are like to those of Esau.'
HaRav Yosef Mizrahi Shelit"a says non-Jews have olam haba if they keep the seven laws.
My daughter came home from school one day to ask my wife whether my parents (who aren't Jewish) would go to heaven when they die. She happened to see Rabbi Kalman Winter, z'tl, the next day, and posed that question. He said: "You tell your daughter that although her grandparents are not Jewish, there is a place for them in Haolam Haba, especially for the merits her father did since he became Jewish, and for the mitzvos that she does in her life."
Yes, the Meiri in Beis HaBechirah (Masechet Sanhedrin 47a) says that every gentile have a share in the World to come as long he/she follows the 7 laws.
Rashi in Parshas Noach regarding the punishment of the Dor Haflaga, Chapter 11, verse 9:
"...and from there… scattered them": This teaches [us] that they have no share in the world to come (Mishnah Sanh. 107b).
Which implies that otherwise they would get Olam Haba, and they weren't Jewish.