PMA Questions

I wonder if anyone can answer a few questions I have regarding PMAed
parts? Suppose I have a part I wish to install on a certified aircraft.
Are the following statements true or false?

1/ If the part is PMAed and the airplane is in the eligibility list,
and the PMA was obtained by virtue of the part being identical to the
original TC holder's design, the part can be installed by an A&P with
a logbook entry.

2/ If the part is PMAed by virtue of an STC which includes the airplane
in its approved model list, the part can be installed, but it must be
documented using a Form 337 which must be signed by an IA.

3/ If the part is PMAed virtue of an STC which does NOT include the
airplane in its approved model list, it MIGHT be possible to get the
part installed, but it must be done via a form 337 and a field
approval.

4/ If the part is not PMAed, it cannot be offered for sale for
installation of a certified aricraft per 21.303 unless it is TSOed, a
standard part etc. It might however be possible to get it installed
using a field approval.

Thanks in advance for any light you can shed on this...