As the scale of data centers continues to grow, there is an increasing demand for interconnection networks to resist malicious attacks. Hence, it is necessary to evaluate the reliability of networks under various fault patterns. The family of generalized $K_4$-hypercubes serve as interconnection networks of data centers, characterized by topological structures with exceptional properties. The $h$-extra edge-connectivity $λ_h$, the $l$-super edge-connectivity $λ^l$, the $l$-average degree edge-connectivity $\overline{λ^l}$, the $l$-embedded edge-connectivity $η_l$ and the cyclic edge-connectivity $λ_c$ are vital parameters to accurately assess the reliability of interconnection networks. Let integer $n\geq3$. This paper obtains the optimal solution of the edge isoperimetric problem and its explicit representation, which offers an upper bound of the $h$-extra edge-connectivity of an $n$-dimensional $K_4$-hypercube $H_n^4$. As an application, we presents $λ_h(H_n^4)$ for $1\leq h\leq 2^{\lceil n/2 \rceil }$. Moreover, for $2^{\lceil n/2\rceil+t}-g_t \le h\le2^{\lceil n/2\rceil+t}$, $g_t=\lceil(2^{2t+2+γ})/3\rceil$, $0\leq t \leq\lfloor n/2\rfloor-1 $, $γ=0$ for even $n$ and $γ=1$ for odd $n$, $λ_h(H_n^4)$ is a constant $(\lfloor n/2\rfloor-t)2^{\lceil n/2\rceil+t}$. The above lower and upper bounds of the integer $h$ are both sharp. Furthermore, $λ^l(H_n^4)$, $\overline{λ^l}(H_n^4)$, $λ_{2^l}(H_n^4)$, and $η_l(H_n^4)$ share a common value $(n-l)2^l$ for $2\leq l\leq n-1$, and we determines the values of $λ_c(H_n^4)$.