Distinct Pairwise Sums

For a set $A = \{a_1, a_2, \dots, a_n\}$ of positive reals ($n \geqslant 2$), define

If $A + A$ has exactly $\dfrac{n(n-1)}{2}$ elements, we say $A$ has property $\Omega$.

1. Determine whether $A_1 = \{1, 2, 4\}$ and $A_2 = \{1, 2, 4, 5\}$ have property $\Omega$. 2. Suppose $B = \{1, 3, p, q\}$ (with $p, q \in \mathbb{N}$ and $3 < p < q$) has property $\Omega$, and the elements of $B + B$ can be arranged into an arithmetic sequence. Find $p$ and $q$. 3. Suppose $A$ has property $\Omega$ and the elements of $A + A$ can be arranged into an arithmetic sequence. Is there a maximum possible number of elements of $A$? If so, find it; if not, explain why.