As far as I know, the transformation of a feasibility problem into an optimization problem has no special name.
As for your second question, the short answer is yes. An optimal solution is a feasible solution which happens to give you the least value (in the case of minimization) of your objective function.
I offer a brief explanation of what is behind what you want to do. An optimization problem can be transformed into an equivalent problem for algorithmic purposes or just to get an explicit solution.
Consider the general convex program
\begin{align}
\min_x \ & \ f(x)\\
\text{s.t.} \ & \ g(x) \leq 0 \\
\ & \ h(x) =0,
\end{align}
then we can use the epigraph transformation, which gives the equivalent problem:
\begin{align}
\min_{x,t} \ & \ t\\
\text{s.t.} \ & \ f(x) \leq t \\
\ & \ g(x) \leq 0 \\
\ & \ h(x) =0.
\end{align}
This transformation preserves convexity.
(The epigraph of a function $f:\mathbb{R}^n \to \mathbb{R}$ is the set of points lying above its graph, i.e.,
$\text{epi}f = \{ (x,\mu) : x \in \mathbb{R}^n, \ \mu \in \mathbb{R}, \ f(x) \leq \mu \})$
In your case, as both $f_1$ and $f_2$ are convex, you can write your feasibility problem as the following optimization problem:
\begin{align}
\min_{x, \delta} \ & \ \delta \\
\text{s.t.} \ & \ f_1(x) \leq \delta \\
\ & \ f_2(x) \leq \delta \\
\ & \ 0 \leq \delta
\end{align}
where the last constraint is added to ensure that your problem is bounded (does not go to $- \infty$).
I hope you find this helpful.
Best Answer
It holds on generic examples, but at the same time it is easy to find failures. Here shown via YALMIP in MATLAB