For example: Can you still get War Suns if you just researched the Red Techs, but none of the Yellows?
Focused Research says "Spend six resources to ignore one prerequisite Technology requirement in the tech tree."
The important part is "in the tech tree." Because the FAQ says: "Focused Research allows you to ignore one prerequisite on the tech tree. Thus, you could not use this card to purchase Advanced Fighters as your first technology."
This means that you always have to think of a prerequisite in the tech tree when using this action card. Let's take your example with the War Sun technology. The card says: "Requires: Deep Space Cannon AND Sarween Tools."
If you think of the tech tree the War Sun technology requires following technologies: Hylar V Assault Laser -> Deep Space Cannon AND Enviro Compensator -> Sarween Tools.
With the action card Focused Research you are allowed to "ignore one prerequisite on the tech tree." So you have to already own either: Hylar V Assault Laser, Deep Space Cannon and Envire Compensator or Hylar V Assault Laser, Enviro Compensator and Sarween Tools to research the War Sun technology with the action card Focused Research because this action card says "on the tech tree."
"Inheritance Systems" is not (yet?) clarified, just it's wording regarding the Strategy Card was corrected.
The Inheritace Systems card says (clarified by the FAQ): "When researching Technology using the Technology Strategy Card, you may spend 2 additional resources to ignoer the Technology's prerequisites."
Since prerequisites is the plural it allows you to ignore all of the prerequisites of the technology. It doesn't matter if you look at the tech tree or at the actual card (since the FAQ says: "The only prerequisites for a technology are the ones written on its Tech card." By the way: This does not count for the Focused Research action card since it clearly states: "on the tech tree.") because either way you ignore all of them. This means that the L1z1x are allowed to research any technology, if they do not own the prerequisites the costs are increased by 2, "spend 2 additional resources to ignore".
This means that it doesn't matter if you use the primary or secondary ability of the Technology strategy card, either way you are using the Technology card and so you can use the Inheritance Systems card.
My question would be: How does Neural Computing interact with IS?
The Neural Computing card says: "Your cost to purchase new Technologies is reduced by 2."
This means that whenever you purchase any technology and you have to pay some amount of resources, this amount is reduced by 2.
So if you are using the Technology I strategy card and use the primary ability and the Inheritance Systems card to skip some prerequisites you would pay 0 to do so. Because the total cost of purchasing a technology is reduced by 2 by the Neural Computing. The total cost was 0+2 due to the Inheritance Systems card.
However, if you are using the secondary ability you have a total cost of 8+2-2 = 8 if you are using the Inheritance Systems card, 8-2 = 6 if you don't use it and 8-X-2 if you have X planets with a technology specialty matching the technology color you are researching. (8+2-X-2 with the Inheritance Systems card).
That is at least what I think after reading all the cards and the FAQ, I just took the exact meaning of the cards and this is what I have got.