1
Pinocchio asked his Rav about Shabbos and Yom Tov, and was told that right before Shabbos comes in, he should say out loud : "I will not lie on this coming Shabbos!"
If Pinocchio never lies on that Shabbos, then all is well. However, if he does lie, then the growth of his nose may be attributed to the fact that his initial statement (now being a lie) made before Shabbos is what triggered the growth of his nose (and not the lie he actually uttered on Shabbos). So his performance of the melachos of Zoreah and Boneh was done while it was still the weekday, and therefore he is exempt from having desecrated Shabbos or Yom Tov.
Although, the lie he uttered on Shabbos has the power to cause the growth all by itself, the previous statement made erev Shabbos is certainly a full cause as well, and therefore it becomes the classic case of "Zeh v' Zeh Gorem", and Pinocchio cannot be held liable.
2
Since Pinocchio is a puppet; then he has the "Din" of an "eved" (slave) since by definition a puppet (even one with sentience and choice) is still a kind of slave.
Slaves (according to the Gemara) are compared Halachically to Karka (land). Therefore the severed nose of Pinnochio has the "Din" of "gedulei karka" and is Kosher for "schach" as a Sukkah roof.
However, if Pinnochio becomes a real boy, then the Sukkah is Pasul (as the nose no longer has the status of "gedulei karka" since Pinnochio is now both human and free. This transformation is "chal Lemafreyah" retroactively, so the nose was never "gedulei karka" in the first place.)
3
Since Pinnochio has some life even in puppet form, then the attached nose is not a "kli", but is "batel" to the living body of Pinnochio. Once it is severed, it still has no "bais kibel" as Pinnochio has no nostrils. So at worst, his nose falls under "peshutei kli eytz" or "smooth wooden objects" which are not mekabel tuma'ah.
Still, some Poskim say that if the lies which fueled this nose were "simple" lies, then the nose is called "pashut kli eytz". But complicated lies ruin that "pashut" status (and then the nose is mekabel tuma'ah.)
Still other Poskim say that all lies are complicated and therefore the nose is always mekabel tuma'ah anyway and therefore many avoid using a mat of Pinnochio noses as "schach".
4
Pinnochio's growth of nose has the same "din" as Tzaraas for Pinnochio and therefore there is an Issur (prohibition) to remove it.
This is learned out because the Metzorah needs to cover himself up to his lip and call out "Tamei" wherever he goes. This covering up to his lip (moustache area) obviously points to the nose.
Levit. 12:45: "...he shall cover himself down to his mustache and call out, "Unclean! Unclean!"
5
From Genesis 2:7, we see that even when a man is created from the inanimate ground, its nose is still imbued with life. Therefore, Pinnochio's nose could never be an Av HaTumah.
"And Hashem Elokim formed man of dust from the ground, and He breathed into his nostrils the soul of life, and man became a living soul."
6
Pinochio's nose is in fact suitable for use in the Mishkan. We learn this out because even though lying is a sin, still there is Biblical example where sinful offerings from pretenders must be used in the Mishkan to set a rememberance and example.
In Bamidbar 17 - 18, … 250 false Kohanim Gedolim (of Korach's rebellion) tried to offer incense and were consumed by fire before Hashem for their falsehood. Yet, their firepans which were made for incense offerings (associated with the NOSE!) were made part of the Mizbeach.
7
אֱמֶת מֵאֶ֣רֶץ תִּצְמָ֑ח
Tehillim 85: 12-13: "Truth will sprout from the earth, and righteousness will look down from heaven. Hashem too will give good, and our land will give its produce."
From here we learn a "Drasha", that we are only "noheg" the laws of orlah and shmittah regarding truthful growth of produce. This excludes Pinnochio's nose which was sprouted from lies.
8
By the Flood it says: Gen. 6:11 "Now the earth was corrupt before Hashem, and the earth became full of robbery."
Corruption and dishonest gain are related to the concept of lies.
Gen. 7:22 "Everything that had the breath of the spirit of life in its nostrils, of all that were on the dry land, died."
So the flood punished through the taking of life which is in the nose.
so we see that only dishonesty is punished through the nose. However, other sins like Lashon Hara etc. that the OP listed, would be punished in other ways.
However,
Genesis 27:27 "And he came closer, and he kissed him, and he smelled the fragrance of his garments, and he blessed him, and he said, "Behold, the fragrance of my son is like the fragrance of a field, which the Lord has blessed!"
When Yaakov pretends to be Esav, he is doing so in essence for Shalom Bayis (his mother told him to do so because the Brachos of Klal Yisroel should not be destroyed etc.)
So we see that pretending for Shalom, is rewarded with the smell of Gan Eden which is associated with the nose.
So the nose will not grow due to Shalom Bayis!
And that verse is 27 of 27.
2+7 = 9 and 2+7 = 9
Emes (truth) = 441, and 4+4+1 = 9 !
