Is the union of $\emptyset$ with another set, $A$ say, disjoint? Even though $\emptyset \subseteq A$? Is the union of $\emptyset$ with another set, $A$ say, disjoint?  Even though $\emptyset \subseteq A$?
I would say, yes - vacuously.  But some confirmation would be great.
 A: Two sets $A, B$ are disjoint iff $A \cap B = \emptyset$.
You have $A \cap \emptyset = \emptyset$. Therefore $A$ and $\emptyset$ are disjoint.
A: It might depend on what you mean by disjoint. I would say that the following definition is reasonable.

Definition. Sets $A$ and $B$ are disjoint if $A \cap B = \emptyset$.

The set $B = \emptyset$ satisfies this, so $A$ and $\emptyset$ are disjoint. But I would not say that this is true vacuously.
A: There is no such thing as a disjoint union of two sets.
Two sets are disjoint when their intersection is disjoint.
By thus definition, the empty set and any set are disjoint.
Usually disjoint is limited to not empty sets.  
A collection K of sets is collectively disjoint when $\cap$K is empty.
A useful notion of a disjoint collection is pairwise disjoint.
A collection K of sets is pairwise disjoint when $\cap$K is empty.
If K is pairwise disjoint, so is K $\cup$ {empty set}.
Again for this notion, the empty set is usually discarded.  
Exercise.  Present a collection of sets that's collectively disjoint but not pairwise disjoint. 
