You are effectively asking the following question:
Consider a metric space $(X,d)$ where $X=\{A_0, A_1, A_2, A_3\}$ is a four-point set. What are the necessary and sufficient conditions for existence of an isometric embedding of $(X,d)$ in the 3-dimensional Euclidean space $E^3$?
Here, $A_1, A_2, A_3$ correspond to the centers $p_1, p_2, p_3$ of the spheres in your question and $A_0$ corresponds to a triple intersection point of these spheres. Let $d_{ij}=d(A_i, A_j)$, $0\le i, j\le 3$. In your question, $r_1=d_{01}, r_2=d_{02}, r_3=d_{03}$ are the radii of the spheres with centers $A_1, A_2, A_3$. Of course, $d_{ii}=0$ for all $i=0,...,3$ and $d_{ij}=d_{ji}$. Also, we have to have the triangle inequalities
$$
d_{ik}\le d_{ij} + d_{jk}
$$
for all $0\le i, j, k\le 3$. (You already wrote these in your question.) The last necessary and sufficient condition for the existence of an isometric embedding comes from the Cayley-Menger determinant
$$
D=\left| \begin{array}{ccccc}
d^2_{00} & d^2_{01} & d^2_{02} & d^2_{03} & 1\\
d^2_{10} & d^2_{11} & d^2_{12} & d^2_{13} & 1\\
d^2_{20} & d^2_{21} & d^2_{22} & d^2_{23} & 1\\
d^2_{30} & d^2_{31} & d^2_{32} & d^2_{33} & 1\\
1 & 1 & 1& 1 & 0\end{array}
\right|=\left| \begin{array}{ccccc}
0 & r^2_1 & r^2_2 & r^2_3 & 1\\
r^2_{1} & 0 & d^2_{12} & d^2_{13} & 1\\
r^2_{2} & d^2_{21} & 0 & d^2_{23} & 1\\
r^2_{3} & d^2_{31} & d^2_{32} & 0 & 1\\
1 & 1 & 1& 1 & 0\end{array}
\right|.
$$
Namely, $D\ge 0$.
For details, see my answer here.
