See answers below
Questions for Pericles:
1) During battles when you have eliminated all the opposition's land and naval troops and you are left with enemy bases does it cost 1 or 2 points to eliminate one single base ?
2) Do you get any honor when you build a base using Diplomacy ?
You always get honor for building/ converting bases.
3) The black Persian bases, do you get any honor when building it with a League Issue ? (I doubt it as I'm sure I read somewhere that Persia is not classified as a Theater even).
No exceptions, if a League issue builds a base you get the honor benefit.
4) One for me finally, regarding the 3 BRAIN TRUST cards that are linked to the King. When required to select one do you have to select that one out of the 3 totally RANDOMLY ??
during the assembly phase ?
Also if you do use one of the 3 brain trust cards and you cannot use any for the NEXT ROUND (have to have the two left face up) at the END of that NEXT ROUND are you obliged to throw the two back into the discard pile and receive the usual 9? (I know though at the START of the NEXT round you only receive 6 cards).
All unused brain trust cards are shuffled into the next turn, but the player who used his brain trust is only dealt 6, not 9, cards that turn. In essence if you use your brain trust on turn 2, all brain trust cards are shuffled in the turn 3 assembly but that player is only dealt 6 cards and has to use all of them with their leader to make up their 7 card hand.
I hope that helps,