The easiest strategy here is to realize that SDP solvers solve SDPs defined by data $(b,C,A)$ defining the primal and dual pair, and they solve both simultaneously.
The primal (which you are trying to fit your model to)
\begin{align*}
\text{minimize} \quad & C \bullet X \\
\text{s.t} \quad & A_i \bullet X = b_i \quad i = 1, \ldots, m \\
& X \succeq 0 \\
\end{align*}
and the dual
\begin{align*}
\text{maximize} \quad & b^Ty \\
\text{s.t} \quad & C - \sum_{i=1}^m A_i y_i \succeq 0 \\
\end{align*}
The problem you are trying to solve is easily interpreted from the dual side
\begin{align*}
\text{maximize} \quad & -t \\
\text{s.t} \quad & -A - (-I)t \succeq 0 \\
\end{align*}
I.e you already have data to send to the solver $(-1,-A,-I)$. The dual solution returned by the solver is your $t$.
Brian Borchers' comment addresses the first question, pointing out that Slater's condition requires strict feasibility in all inequalities.
For the second question, the KKT conditions are more than just $X \bullet (C - \sum y_i A_i) = 0$ (stationarity) alone. Wikipedia additionally lists (adapting the general conditions to the case of SDPs, with your notation and sign conventions)
- Primal feasibility: $d_i(X) = b_i - A_i \bullet X \geq 0$, $X \succeq 0$
- Dual feasibility: $y_i \leq 0$, $C - \sum y_i A_i \succeq 0$
- Complementary slackness: $\sum_i y_i d_i(X) = 0$
These, together with stationarity and a constraint qualification (such as Slater's condition), are jointly necessary conditions fulfilled by all optima.
Your proof of $(X, y_i)\text{ optimal }\Rightarrow \mathrm{KKT}(X, y_i)$ (KKT necessary) shows how optimality implies primal and dual feasibility at $X$, and uses it to prove complementary slackness, and then stationarity.
For $\mathrm{KKT}(X, y_i) \Rightarrow (X, y_i)\text{ optimal}$ (KKT sufficient), as you have noted, you need complementary slackness to prove the stationarity of $(X', y_i')$ from the stationarity of $(X, y_i)$. Since stationarity of $(X', y_i')$ alone is sufficient for its equality-constrained problem, whereas inequality-constrained problems require all KKT conditions to be fulfilled, it is not surprising that fulfilling some of the KKT conditions for $(X, y_i)$ does not imply fulfilling the condition for $(X', y_i')$.
As a side note, even for problems with only equality constraints, stationarity and primal feasibility are jointly necessary.
Best Answer
Primal-dual interior point methods can solve this formulation efficiently and there are multiple open source codes available to do so. CSDP can solve it well and is documented here. If you use that paper, I'd ignore the operator $B$ to make things simpler. Alternatively, SDPT3 is also an excellent code and is documented here. That said, from a pedagogical point of view, these papers are a terrible place to start. In my opinion, a much better path is to start from Wright's book Primal-Dual Interior-Point Methods, which covers the algorithm for linear, not semidefinite, cones. Really, most of the algorithm is the same, but the primal and dual variables are no longer vectors, but matrices. In Wright's book, this means that $X$ and $S$ are matrices and no longer $Diag(x)$ and $Diag(s)$ respectively. This also adds the complication of symmetrization because the optimization steps will in general not be symmetric, but the CSDP and SDPT3 papers will cover what to do in this situation.
To be sure, I am aware that there is continued research on SDP max-cut relaxations and most likely algorithms, so it's possible that there is better than what I've suggested above. That said, those algorithms are really good and are a reasonable place to start.