I do not know if there is a way to get the isoperimetric inequality from the spectral gap, but both can be proven in almost the same way. The classical references for the linear isoperimetric inequality are S.-T. Yau, "Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold", Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 487–507 and
Yurii D. Burago and Victor A. Zalgaller, "Geometric inequalities".
I like this proof so let me give it here (this is Burago-Zalgaller presentation). For any unit tangent vector $u$ and positive real $r$, let $s(u,r)$ be the "candle function" defined by
$$dy = s(u,r) \,du \,dr$$
when $y=\exp_x(ru)$ and $u\in UT_xM$. Up to a normalization, this is simply the jacobian of the exponential map. The curvature hypothesis implies $(\log s(u,r))'\geqslant \sqrt{-\kappa}(n-1)$ where the prime denotes derivative with respect to $r$ (this is a consequence of Günther's inequality).
$\Omega$ is contained in the union of all geodesic rays from any fixed point $x_0$ to $\partial \Omega$. Let $U\subset UT_{x_0}M$ be the set of unit vectors generating geodesics that intersect $\Omega$, and for $u\in U$ let $r_u$ be the last intersection time of the geodesic generated by $u$ with $\Omega$. Then
$$\mathrm{Vol}(\partial \Omega) \geqslant \int_U s(u,r_u) \,du$$
and
$$\mathrm{Vol}(\Omega) \leqslant \int_U \int_0^{r_u} s(u,t) \,dt\,du.$$
Now, writing $s(u,r_u)=\int_0^{r_u} s'(u,t) \,dt$ and using Günther's inequality, the desired result comes.
I cannot help self-advertising: in fact, the same conclusions (Günther inequality, hence both the linear isoperimetric inequality and the spectral gap of MCKean) hold under a weaker curvature bound (some higher but non-positive sectional curvature can be compensated by enough more negative sectional curvature in other directions). This is explained in an arXiv paper with Greg Kuperberg, "A refinement of Günther's candle inequality" [arXiv:1204.3943].
I think you're inadvertently opening a big can of worms. The question can be answered by a combination of two facts: the absence of branch points in (almost-)minimising hypersurfaces and Allard's regularity theorem.
Specifically, the tangent cones to $H$ at $h$ must be multiples of an $n$-dimensional hyperplane $P$ say, with some multiplicity $Q \in \mathbf{Z}_{>0}$. The tangent cones cannot be more complicated minimal cones, as for example a Frankel-type argument demonstrates. Let $$ \mathbf{C} \in \mathrm{VarTan}(H,h)$$ be a tangent cone to $H$ at $h$: this is a (singular) minimal surface. Knowing that $\mathbf{C}$ is a stationary varifold is enough for now. By construction the cone is supported in a closed half-space, for example $$\mathrm{spt} \, \mathbf{C} \subset \{ X \in \mathbf{R}^{n+1} \mid X^{n+1} \geq 0 \}.$$
The intersection of $\mathbf{C}$ with the unit sphere $\partial B$ defines a stationary varifold contained in a hemisphere. Now on the one hand, as $\partial B$ has positive Ricci curvature, $\mathrm{spt} \, \mathbf{C}$ and $\partial B \cap \{ X^{n+1} = 0 \}$ must intersect: this is Frankel's theorem. On the other hand, this intersection must be tangential, and the maximum principle forces them to coincide: $$\mathrm{spt} \, \mathbf{C} = \{ X^{n+1} = 0 \}.$$
Therefore letting $P = \{ X^{n+1} = 0 \}$ one has $$\mathbf{C} = Q \lvert P \rvert.$$
When $H$ is minimising (or almost-minimising), then it cannot have branch point singularities, and necessarily $Q = 1$.
Therefore the tangent cones are multiplicity one tangent planes, and by Allard regularity $H$ must be smooth in a neighbourhood of the point $h$.
Best Answer
I guess Gromov wanted to say that there is a lower bound for $\mathop{\rm vol}\partial V_0$ in terms of $\mathop{\rm vol} V_0/\mathop{\rm Vol} V$, $\mathop{\rm diam}V$ and lower bound for Ricci curvature. The same proof as in "Paul Levy's Isoperimetric Inequality", gives such a bound, but it is not longer sharp.
BTW, there is an analog of Levy--Gromov for open manifolds with $\mathop{\rm Ricc}\ge 0$. It is sharp and gives a lower bound for $\mathop{\rm vol}\partial V_0$ in terms of $\mathop{\rm vol} V_0$ and the volume growth of $V$, BUT as far as I know it is not written. (Please correct me if I am wrong.)