If צחק refers to the 3 worst misdeeds why Itzchak was called so?

Question

On the Posuk: "וַתֵּרֶא שָׂרָה אֶת־בֶּן־הָגָר הַמִּצְרִית אֲשֶׁר־יָלְדָה לְאַבְרָהָם מְצַחֵק׃" Rashi (Gen 21.9) explains the meaning of the root צחק:

מצחק: לְשׁוֹן עֲ"זָ, כְּמוֹ שֶׁנֶּאֱמַר וַיָּקֻמוּ לְצַחֵק ,
דָּ"אַ לְשׁוֹן גִּלּוּי עֲרָיוֹת, כְּמָה דְּתֵימָא לְצַחֶק בִּי,
דָּ"אַ לְשׁוֹן רְצִיחָה, כְּמוֹ יָקוּמוּ נָא הַנְּעָרִים וִישַׂחֲקוּ לְפָנֵינוּ וְגוֹ'

This means worshipping idols,
... Another explanation is that it refers to immoral conduct,
...Another explanation is that it refers to a murder

If that's the meaning of the root, how our forefather could be named Itzchak?

Answer 1

The name יצחק means he will rejoice, meaning that this will be shown to be good. The term used for Yishmael, מצחק, means that he mocked actively and viciously. It therefore is a completely different usage.

Rav Hirsch explains that their are two different meanings involved. Vayeirah 21:3

3 And Abraham named his son who had been born to him, whom Sarah had borne to him, Isaac.

the birth of this child was צחק, something, according to natural conditions, rather laughable (see above on Chapter XVII, 17). So when Abraham called his son יצחק these facts were vividly in his mind.

The statement in Vayeira 17:17 shows:

17 And Abraham fell on his face and rejoiced, and he said to himself, "Will [a child] be born to one who is a hundred years old, and will Sarah, who is ninety years old, give birth?"

On the other hand, Yishmael actually went to great extremes in his making fun of the ideas of Avraham. As Rav Hirsch says on Vayeira 21:9:

He took in just as much of the great ideas of Abraham to make him ironically disdain them, and over that which the world in general greeted merely with צחוק, in which (V.6) derision was only lightly mixed, Ishmael was completely מְצַחֵק, and was therefore completely unfit to join with Isaac in the heritage of Abraham.

Answer 2

And I say: (Rabbi Shimon ben Yochai) “God forbid that there be, in the house of this righteous man, anyone doing such things! Rather, this “playing around” has to do with the inheritance - Ishmael was fooling around and saying, “I am the eldest, and so I will take a double portion.”
https://www.sefaria.org/Tosefta_Sotah.6?lang=bi

in this context, perhaps one can say that Ishmael was doing more than just playing around. He was, more precisely, “Isaac-ing.” In other words, he was trying to be Isaac - to take his place, to be the inheritor.