To start, I would suggest a read through 14 CFR Part 21.
But to summarize: a TC is necessary to manufacture a new aircraft, so TCs are the property of the manufacturer of the aircraft. The TC confers the following privileges:
The holder or licensee of a type certificate for a product may—
(a) In the case of aircraft, upon compliance with §§21.173 through
21.189, obtain airworthiness certificates;
(b) In the case of aircraft engines or propellers, obtain approval for
installation on certificated aircraft;
(c) In the case of any product, upon compliance with subpart G of this
part, obtain a production certificate for the type certificated
(d) Obtain approval of replacement parts for that product.
[Doc. No. 5085, 29 FR 14564, Oct. 24, 1964, as amended by Amdt. 21-92,
74 FR 53386, Oct. 16, 2009]
OTOH, An STC can be obtained by anyone including the TC holder. The big difference is that it is a 'supplement' to a TC. That is, it modifies the design specified by the TC.
So we can look to the following section of Part 21:
§21.113 Requirement for supplemental type certificate.
(a) If a person holds the TC for a product and alters that product by
introducing a major change in type design that does not require an
application for a new TC under §21.19, that person must apply to the
FAA either for an STC, or to amend the original type certificate under
subpart D of this part.
(b) If a person does not hold the TC for a product and alters that
product by introducing a major change in type design that does not
require an application for a new TC under §21.19, that person must
apply to the FAA for an STC.
(c) The application for an STC must be made in the form and manner
prescribed by the FAA.
[Doc. No. FAA-2006-25877, Amdt. 21-92, 74 FR 53387, Oct. 16, 2009;
Doc. No. FAA-2018-0119, Amdt. 21-101, 83 FR 9169, Mar. 5, 2018]
As stated here, an STC applies to any major modification that doesn't require an new TC under §21.19 which states:
§21.19 Changes requiring a new type certificate.
Each person who proposes to change a product must apply for a new type
certificate if the FAA finds that the proposed change in design,
power, thrust, or weight is so extensive that a substantially complete
investigation of compliance with the applicable regulations is
[Doc. No. 28903, 65 FR 36265, June 7, 2000]
So your question as to TC vs. STC really only applies to the manufacturer. And as stated here, it's at the discretion of the FAA. But as a point of reference, every Boeing 737 ever built has been built under TC Data Sheet A16WE (currently Rev 64).
Typically manufacturers will continue to modify their TC to support continuing production. They will use an STC when the changes are to modify previously manufactured aircraft.
Third parties that modify aircraft will also get STCs. A large percentage of STCs are obtained by third parties. The STCs only cover the changes from the aircraft's existing certification basis (which is the original TC plus any previously installed STCs.) An STC may be for a single aircraft, a single aircraft type, or for multiple types (an "Approved Model List STC" or AML STC). The last is common for general aviation avionics.
So back to your original list of mods. All of them are major mods and would require at least an STC, though the last one would likely require an modification of the TC if not a new one. Replacing the entire fuselage is a huge change. If Piaggio wanted to make those changes to their design going forward, they would likely modify their existing TC. If the FAA decreed the modification was extreme enough to warrant a new TC, they would have to apply for a new TC.
If someone other than Piaggio wants to make the mods, they would have to get an STC and use it to modify an aircraft that was manufactured by Piaggio. Once the STC is obtained, the STC holder has the following privileges:
The holder of a supplemental type certificate may—
(a) In the case of aircraft, obtain airworthiness certificates;
(b) In the case of other products, obtain approval for installation on
certificated aircraft; and
(c) Obtain a production certificate in accordance with the
requirements of subpart G of this part for the change in the type
design approved by the supplemental type certificate.
[Doc. No. 5085, 29 FR 14568, Oct. 24, 1964, as amended by Amdt. 21-92,
74 FR 53387, Oct. 16, 2009]
One advantage to an STC, is that the owner can sell the rights to use the STC to another. So the following 'permission' typically involves the exchange of money:
§21.120 Responsibility of supplemental type certificate holders to provide written permission for alterations.
A supplemental type certificate holder who allows a person to use the
supplemental type certificate to alter an aircraft, aircraft engine,
or propeller must provide that person with written permission
acceptable to the FAA.
[Doc. No. FAA-2003-14825, 71 FR 52258, Sept. 1, 2006]