I will add a few comments to Bjørn Kjos-Hanssen's answer.

1. Does property (A) have a name in the literature? Is it a studied notion?

The property is a kind of independence property. A set-theorist's antichain $S$ of a lattice $L$ is called weakly independent if whenever $a_1,\ldots,a_{n+1}\in S$ are distinct, then $(a_1\vee\cdots\vee a_n)\wedge a_{n+1} = 0$. It is called strongly independent if whenever $I$ and $J$ are finite sets of indices, then $(\bigvee_{i\in I} a_i)\wedge (\bigvee_{j\in I} a_j)=(\bigvee_{k\in I\cap J} a_k).$ Property (A) lies between these two concepts.

2. When does (A) coincide with being an antichain? Only in distributive lattices?

It coincides in any pseudocomplemented lattice. These need not be distributive. For example, the congruence lattice of a semilattice is pseudocomplemented and almost never distributive. The order-dual of the lattice of convex subsets of a finite chain is pseudocomplemented, but not distributive when the chain has more than 2 elements.

3. Does maximal with respect to (A) imply being a maximal antichain?

I would guess that being maximal with respect to (A) rarely means being a maximal antichain for lattices that are not pseudocomplemented. For a simple example, if $S$ is an antichain in the lattice $L$ of subspaces of a vector space $V$, and $S$ satisfies Property (A), then the size of $S$ cannot exceed the dimension of $V$. But $L$ can have antichains larger than the dimension of $V$. For example, if $V$ is finite dimensional and $T\subseteq L$ is the set of 1-dimensional subspaces, then $T$ is larger than any antichain in $L$ that has Property (A).

I will add a few comments to Bjørn Kjos-Hanssen's answer.

1. Does property (A) have a name in the literature? Is it a studied notion?

The property is a kind of independence property. A set-theorist's antichain $S$ of a lattice $L$ is called weakly independent if whenever $a_1,\ldots,a_{n+1}\in S$ are distinct, then $(a_1\vee\cdots\vee a_n)\wedge a_{n+1} = 0$. It is called strongly independent if whenever $I$ and $J$ are finite sets of indices, then $(\bigvee_{i\in I} a_i)\wedge (\bigvee_{j\in I} a_j)=(\bigvee_{k\in I\cap J} a_k).$ Property (A) lies between these two concepts. In a modular lattice all three notions agree (weak independence, Property (A), and strong independence).

2. When does (A) coincide with being an antichain? Only in distributive lattices?

It coincides in any pseudocomplemented lattice. These need not be distributive. For example, the congruence lattice of a semilattice is pseudocomplemented and almost never distributive. The order-dual of the lattice of convex subsets of a finite chain is pseudocomplemented, but not distributive when the chain has more than 2 elements.

3. Does maximal with respect to (A) imply being a maximal antichain?

I would guess that being maximal with respect to (A) rarely means being a maximal antichain for lattices that are not pseudocomplemented. For a simple example, if $S$ is an antichain in the lattice $L$ of subspaces of a vector space $V$, and $S$ satisfies Property (A), then the size of $S$ cannot exceed the dimension of $V$. But $L$ can have antichains larger than the dimension of $V$. For example, if $V$ has finite dimension greater than one and $T\subseteq L$ is the set of 1-dimensional subspaces, then $T$ is larger than any antichain in $L$ that has Property (A).