At what point does an aircraft need a new type certificate vs a STC?
Since this is an aircraft for personal use, is under 19,000lbs gross weight, and less than 19 seats it will fall under Part 23.

If you don't plan on selling "kits" to do these modifications, then an STC is not required. STC's are only for modifications intended for sale to aircraft for installation by an A&amp;P mechanic. STC's are required not for the kit itself, but for each aircraft type (and sometimes variant) that it is to be installed on. This is why the STC process can be very expensive. 

You would not need an entirely new type certificate either, again which would be required if you were doing these modifications as a business for sale as another aircraft type, or if the original manufacturer modified the aircraft in some significant way. 

What you would need in almost all the cases you described above is an FAA Form 337, which describes "major repair &amp; alterations to airframe or power plant". This document will require you (and your A&amp;P) to make detailed descriptions and drawings of the modification and submit it to an FSDO. The FSDO will review the modifications and approve or disapprove of the changes. It is advisable to work with the FSDO from the beginning, not the end.




  So, which of these alterations would require a new TC of the aircraft vs a STC?
  
  
  Lengthening or shortening the fuselage?
  Changing the engine(s) from say piston to turbine or turbine to electric fans?
  Adding extra components to the airframe, like batteries, controllers, etc. to make a hybrid aircraft?
  Changing the wings to newer ones from another certified aircraft (re-winging)?
  Changing the landing gear to a newer model with higher load capabilities?
  Expanding the fuselage diameter by 10%?
  


None of them require a new TC or STC, however your first question would probably be difficult at best. The second question doesn't make a lot of sense for the Piaggio since it already has turbine engines, and there are no electric fans of the size that would be required. Changing the wing or modifying the landing gear are pretty common things, usually in the form of STOL kits or FOD deflectors. Some of them would require extensive testing from the FAA before issuing the airworthiness paperwork.

The last one though will nearly be impossible. It would be a lot cheaper to just buy an aircraft that fit your needs, like the Pilatus PC-24 than try to make that modification.