Is my understanding of the connections between anti-/a-/symmetry and reflexivity in relations correct? A relation is symmetric if when aRb then bRa, antisymmetric if when aRb and bRa then a=b, and asymmetric if when aRb then not bRa. Does it follow that every antisymmetric relation R is also reflexive (Id is a subset of R), and that a relation both antisymmetric and symmetric is equal to the identity relation (Id = R)?

You're correct about all of the definitions, but your conclusions are not true. Let $A$ be our set, and take $R$ to be _any_ subset of $\Delta_A = \\{(a,a) \mid a\in A\\}$. This is a relation which is both symmetric and antisymmetric, but is not reflexive, since there's no reason to suggest that it is all of $\Delta_A$.

As a concrete example, take $A=\\{1,2,3\\}$ and $R=\\{(1,1)\\}$. It is clearly symmetric, and it trivially satisfies anti-symmetry.