If one embarrassed someone and regrets it.... Also if one calls his friend a derogatory nickname...
Rambam, Hilchos Teshuvah 3:14:
וְיֵשׁ עֲבֵרוֹת קַלּוֹת מֵאֵלּוּ וְאַף עַל פִּי כֵן אָמְרוּ חֲכָמִים שֶׁהָרָגִיל בָּהֶן אֵין לָהֶם חֵלֶק לָעוֹלָם הַבָּא וּכְדַאי הֵן לְהִתְרַחֵק מֵהֶן וּלְהִזָּהֵר בָּהֶן. וְאֵלּוּ הֵן. הַמְכַנֶּה שֵׁם לַחֲבֵרוֹ ... וְהַמַּלְבִּין פְּנֵי חֲבֵרוֹ בָּרַבִּים. ... בַּמֶּה דְּבָרִים אֲמוּרִים שֶׁכָּל אֶחָד מֵאֵלּוּ אֵין לוֹ חֵלֶק לָעוֹלָם הַבָּא כְּשֶׁמֵּת בְּלֹא תְּשׁוּבָה אֲבָל אִם שָׁב מֵרִשְׁעוֹ וּמֵת וְהוּא בַּעַל תְּשׁוּבָה הֲרֵי זֶה מִבְּנֵי הָעוֹלָם הַבָּא
There are sins which are lighter than these [mentioned previously, 3:6-13], but nevertheless the Sages said that one who regularly does them does not have a portion in the World to Come, and it is fitting that one should distance himself from them and be careful regarding them. And these are they: One who calls his friend by a nickname [which he doesn’t like - BM 58b] ... one who embarrasses his friend in public... When are these words said, that these people do not have a portion in the World to Come? When he dies without repenting. But if he returns from his wickedness, and he dies having repented, he has a portion in the World to Come.
Combine that with what he says earlier, in 2:9:
אֲבָל עֲבֵרוֹת שֶׁבֵּין אָדָם לַחֲבֵרוֹ כְּגוֹן הַחוֹבֵל אֶת חֲבֵרוֹ אוֹ הַמְקַלֵּל חֲבֵרוֹ אוֹ גּוֹזְלוֹ וְכַיּוֹצֵא בָּהֶן אֵינוֹ נִמְחַל לוֹ לְעוֹלָם עַד שֶׁיִּתֵּן לַחֲבֵרוֹ מַה שֶּׁהוּא חַיָּב לוֹ וִירַצֵּהוּ. אַף עַל פִּי שֶׁהֶחֱזִיר לוֹ מָמוֹן שֶׁהוּא חַיָּב לוֹ צָרִיךְ לְרַצּוֹתוֹ וְלִשְׁאל מִמֶּנּוּ שֶׁיִּמְחל לוֹ. אֲפִלּוּ לֹא הִקְנִיט אֶת חֲבֵרוֹ אֶלָּא בִּדְבָרִים צָרִיךְ לְפַיְּסוֹ וְלִפְגֹּעַ בּוֹ עַד שֶׁיִּמְחל לוֹ.
But sins which are between man and his friend, such as one who wounds his friend, or one who curses his friend, or one who robs him, or the like, he is not forgiven [Above] ever until he he gives to his friend that which he owes him and appeases him. Even if he returns the money which he owes him, he still must appease him and ask for forgiveness from him. Even if he didn’t harm his friend with anything but words, he still must appease him and entreat him until he forgives him.
This latter Rambam is essentially quoting the Mishnah, Yoma 85b.
It should be noted that Tosfos (BM 58b s.v. “Chutz miGimmel”) takes a slightly different view of these sins:
חוץ מג' שיורדין ואין עולין - אין לפרש דאין עולין לעולם ... אלא ה"פ כל היורדין עולין מיד ואין אור של גיהנם שולט בהן ... אבל יש מהם נדונין לעולם המלבין פני חבירו ברבים דאמר בסמוך אין לו חלק לעולם הבא ... אי נמי מלבין נמי נהי דאין לו חלק לעולם הבא אינם נדונין יותר מי"ב חדש אלא לאחר י"ב חדש לא חיין ולא נדונין ושרוים בלא טובה ובלא רעה... וכולהו בשלא עשה תשובה דתשובה מועלת לכל דבר
[When the Gemara says that these three don’t ever leave Gehennom,] don’t explain this to mean that they never leave ... rather, this is its explanation: everyone who descended into Gehennom leaves immediately, and the fire of Gehennom does not rule over them. ... But some of these [mentioned in BM] are judged eternally - one who embarrasses his friend in public, as we say momentarily, does not have a share in the World to Come [while others mentioned are judged for 12 months]. Alternatively, one who embarrasses, also: granted that he doesn’t have a share in the World to Come, he’s not judged longer than 12 months, but after 12 months, he doesn’t live, nor is he judged, and he rests with no good nor bad. ... But all of this is when he doesn’t repent, for repentance helps for anything.
According to the above halachos, one must repent in full, which, for sins between man and his friend, includes asking for full forgiveness. If he doesn’t ask for forgiveness, his repentance is incomplete, and he will be judged, whether like the Rambam, or Tosfos, or whatever Hashem decides is appropriate for him.
What if this happened before Bar Mitzvah...before 20?
Why would you think this is any different than the usual rule, that Divine punishment is only meted out at 20?
What if the degree of embarrassment was minor?
Who are you to judge what the victim feels? You might not think that it’s minor, but his entire world may have been shattered as a result of your careless remark. But let’s say that you’re psychic and you actually know how embarrassed he was by it. Let’s go back to the Gemara everyone is discussing (BM 58b):
מכנה היינו מלבין אע"ג דדש ביה בשמיה
One who calls [his friend by] a nickname - that’s the same as one who embarrasses! [Answer: He’s punished] even if he is used to it being used as his name.
Translation follows Rashi.
So we see from here that regardless of the amount of embarrassment, it still is a sin.
What if the one/s he offended was not nice to him...as well?
Even worse. Now you’re also in violation of “Love your neighbor like yourself.” This is in accordance with the famous Gemara in Shabbos 31a:
אמר לו דעלך סני לחברך לא תעביד זו היא כל התורה כולה ואידך פירושה הוא זיל גמור.
Hillel said to him, “That which you hate when done to you, do not do to your friend. This is the entire Torah. The rest is commentary; go and learn.”
as understood by Maharsha there:
והיינו דכתיב בתורה ואהבת לרעך כמוך
And this is that which is written in the Torah, “you should love your friend like yourself.”
Note that this is not an issue of revenge, as Rashi to Vayikra 19:18 notes, quoting Yoma 23a:
לא תקם. אָמַר לוֹ הַשְׁאִילֵנִי מַגָּלְךָ, אָמַר לוֹ לַאו! לְמָחָר אָמַר לוֹ הַשְׁאִילֵנִי קַרְדּוּמְךָ, אָמַר לוֹ, אֵינִי מַשְׁאִילְךָ כְּדֶרֶךְ שֶׁלֹּא הִשְׁאַלְתָּנִי, זוּ הִיא נְקִימָה;
“Don’t take revenge.” Reuven says to Shimon, “Lend me your sickle,” and Shimon says back, “No.” The next day, Shimon says to Reuven, “Lend me your shovel,” and Reuven says back, “I won’t lend you, just like you didn’t lend me.” This is revenge.
So revenge is not doing something to someone, but rather not doing something to someone.
Regarding your final question, I encourage you to post it separately, as while all of the above questions are somewhat contained, this last one is a completely different can of worms.