There are researches to what level different traits (including educational abilities and character)…
Sceptical Meerkat

Studies of twins have been largely discredited by later research (many were shown to be fraudulent) and studies of adopted children are generally discounted because it is not possible to separate the proposed effect of “inheritance” from experience in the womb and around birth. Add in the science of epigenetics and you find that environment & experience affects genetics by switching on & off individual genes and thereby makes the distinction between “nature” and “nurture” impossible to specify.

The truth is that it is not possible to identify the complex network of genes behind intellectual or emotional characteristics to any significant extent, and any more general investigation boils down to deciding that because children usually resemble their parents, “it must be genetic”… which is a circular argument of the worst kind.

You are correct in saying that I would be very surprised if you could produce any proof of heritability of intelligence, since by definition intelligence is an infinite dimensional fuzzy space, which means it is shown by mathematics to be fundamentally not measurable. And “temperament” is such a vague expression, that too is clearly not something that can be subject to good-quality research.