According to https://wiki.dothraki.org/High_Valyrian_Pronouns#Possessive_pronouns there is apparently no difference :)
Actually, I think that zȳha is for for solar and lunar, but jāha is for terrestrial and aquatic. Since you (understandably) skipped past the personal pronouns bit, there was the answer up there, so I think that the people who wrote that thought they need not write it again.
Not necessarily. The pronoun agrees with its antecedant, and you don't see the antedecent. It doesn't agree with the noun it's modifying! That is, the agreement morphology agrees with the noun its modifying—not the pronoun itself. That's governed only by the antecedent.
In other words, that would be like saying "His daughter is happy" should be "Her daughter is happy" because "daughter" is feminine—even though we're talking about a man's daughter.
@GodofChaos19: No. It's just like "he" and "she". It has nothing to do with the speaker. If I'm talking about a man, I say "his daughter". If I'm talking about a woman, I say "her daughter". It doesn't matter what my sex is, or what the sex of the possessed noun (the daughter) is: It matters what the sex of the possessor is.
This is exactly the same in High Valyrian. If the possessor is Lunar or Solar, use zȳh-; if the possessor is Terrestrial or Aquatic, use jāh-.