I'm having trouble understanding why a conjunction is used in the first, and a conditional in the latter case. For example we have:
Every human is a mammal: $(\forall x)(H(x)\to M(x))$
Some animals are mammals: $(\exists x)(A(x)\land M(x))$
But why can't both be used exchangeably? The argument seems to be about the truth value of the statement when there are no humans/animals in the domain to begin with, but it's not very convincible to me.
Regarding "Every human is a mammal", the traditional analysis of Categorical proposition assumes that "universal terms" (the predicates $A, H, M$) are always instantiated. Thus, "Every human is a mammal" is False only in case that there are some human that is not a mammal, i.e. when $\exists x(Hx \land \lnot Mx)$ is True.
Thus, if $\exists x(Hx \land \lnot Mx)$ is the negation of "Every human is a mammal", we have that this one is symbolized by $\lnot \exists x(Hx \land \lnot Mx)$, which is equivalento to: $\forall x(Hx \to Mx)$.
See also vacuous truth.
Regarding "Some animals are mammals", if we formalize it with: $\exists x (Ax \to Mx)$ we have that if there are no animals, then $Ax$ is False for every possible value of $x$ and thus $Ax \to Mx$ is always True.
This means that $\exists x (Ax \to Mx)$ is True also when there are no animals, contrary to our intuition about the truth value of "Some animals are mammals".