Probability that a random graph will remain planar after adding an edge
Let $S$ be the set of planar graphs on $n$ vertices, and let $E(S,\overline{S})$ be the set of edges between $S$ and its complement, that is, the set of pairs $(G,e)$ where $G \in S$ and $G \cup \{e\} \notin S$. The edge-isoperimetric inequality states that
$$
\frac{|E(S,\overline{S})|}{|S|} \geq \binom{n}{2} - \log_2 |S|.
$$
It follows that the probability that you're after is at most
$$
\frac{\log_2 |S|}{\binom{n}{2}-(3n-6)}.
$$
It is known that $\log_2 |S| = O(n \log n)$ (see for example Gimenez and Noy, The number of planar graphs and properties of random
planar graphs, and so we conclude that your probability is at most $O\left(\frac{\log n}{n}\right)$.

On the other hand, the same paper shows that with constant probability, a random planar graph has an isolated vertex. When this happens, there is $\Omega\left(\frac{1}{n}\right)$ probability to add an edge adjacent to it, leaving the graph planar. Hence your probability is at least $\Omega\left(\frac{1}{n}\right)$.