I think the best way to decide about the rules is to follow the exact wording.
The rules of SotT say: "Each Flagship has its own unique cost, combat value, movement, capacity, and special ability …"
The normal rules of TI say: "Unlike the Dreadnought, however, a War Sun ignores the presence of a PDS unit's planetary shield."
I think that a Flagship is only affected by cards, technologies and whatsoever which say the word Flagship since they are "unique" units, meaning that they do not share anything with other units.
Of course: The abilities which units can have are described in the TI rule book and it may seem as if they define the ability. But the abilities are neutral and some units share the same abilities, meaning that not one unit has the ability of another unit, it means that many units can have the same ability, that's a big difference.
So there are three units which share the ability "Planetay Bombardment": The Dreadnought, War Sun and the Arc Secundus.
So they share the ability to bombard a planet if:
A planetary invasion is about the take place, meaning only if Ground Forces are landing.
There are no PDS on the planet.
Now you may ask: But the War Sun …
The War Sun has the basic ability of Planetary Bombardment but it has the additional ability to ignore PDS and invading Ground Forces. Meaning the War Sun has an additional ability which alters the original ability of Planetary Bombardment which is described in the TI rulebook in the part about the Dreadnought (which does not mean that only the Dreadnought has this original Planetary Bombardment ability, he only shares it and is described before the War Sun is described).
The Arc Secundus has the ability to use Planetary Bombardment, but not any modifications, so it cannot ignore PDS and invading Ground Forces.
About the leaders: As Kouteki has said: "Flagships do not get a penalty …"
However, I think that you may change this rules as you like because the TI rulebook allowed the general to give a penalty to all the units which were able to perform a planetary bombardment at that time. However, the FAQ does not say anything about that.
About the scientist: His ability affects War Suns, not Flagships or Dreadnoughts, so it does not matter because the Arc Secundus can't bombard a planet with a PDS anyway.