How correlated is the chirality of different molecules?
Hypothetically if an alien species were found , and if they contained similar biochemical polymers (DNA with A,T,C,G , Protein with 20 Amino acids etc) then we MAY be able to conclude that they share an origin with us. (Just similar chirality does not necessitates the theory of same origin)

Does the chirality of a single compound more or less determine the chirality of all other compounds in an entire biochemistry?

Yes. As Chiral molecules are almost always produced/purified from chiral medium (containing other chiral molecules and/or chiral physical attributes e.g. polarized light).
Therefore (Assuming that the reaction is enantioselective and you want pure chemical with a specific stereochemistry) the configuration of the chiral center of a new molecule does depend on the reagent chirality. Still this relation is not constant and differs from one reaction to another. For instance textbook SN2 reaction leads to inversion of configuration while Williamson ether synthesis reaction maintains the chirality.
A non-racemic mixture of a chiral compound is resulted from previous non-racemic mixture and so on. therefore the stereochemistry of the first non-racemic mixture determined the S/R , L/R ,(-/+) chirality of the entire life per se whose identity may never be fully understood.