Watched your latest episodes and some answers to some of your questions and a tip.
You didn't need to delete that design you created as a refit for the Lillian class. You just missed a button called "New Armour" at the middle bottom of the screen. This put the newest form of armour you have to the design.
The reason you could not build the new Lillian ship in the same yard was because the ship had a smaller mass. I don't know the exact formula but it is 20% of the build cost but that is ONLY if the ship has an equal mass. The more the mass deviates the less likely it is that two ships can be built in the same shipyard. In your case the new Lillian class was about 10% smaller so that is probably too much difference in size (and cost). If you changed the armour but added some armour rows you could probably have built it in the same yard, but that would then be a little pointless to make the new class I presume.
Another tip is to add some navy personnel to your Fleet Headquarter HQ (TF). They will have a survey specialist and will make surveys in Sol system be done faster.
As far as I know the numbers in the system view for surveing are SP (Survey points). Each Geosensor on the ship give you one point per hour modified with the survey trait of the ship commander and then further by the TF cabinet personnel.
Keep up the good work with your let's play, it's fun to watch...