= Solution
Take the normalization
$$
E(\alpha)=\frac12\int_0^1|\dot\alpha(t)|^2\,dt.
$$
The <first variation of geodesic energy> says that the <critical points> on the fixed-endpoint <path space> are exactly the affinely parametrized <geodesics>. The kernel of the <second variation of geodesic energy>, or <Riemannian index form>, is the space of <Jacobi fields> vanishing at both endpoints. Hence a geodesic is a <nondegenerate critical point> exactly when that space is zero, or equivalently when its terminal endpoint is not a <conjugate point> to its initial endpoint along the given geodesic.
The <Morse index theorem> states that the index is the sum of the multiplicities of all <conjugate points> strictly inside the parameter interval:
$$
\boxed{\operatorname{ind}(\gamma)=\sum_{0<t<1}\dim\{J\in\mathcal J_\gamma:J(0)=J(t)=0\}.}
$$
The multiplicity at the terminal endpoint gives nullity rather than an additional contribution to the index. These statements use the usual Sobolev $H^1$ completion of the <path space>; smooth paths have the same homotopy type.
First suppose $p\ne q$ as well as $p\ne -q$, and put $\theta=d(p,q)\in(0,\pi)$. There is a unique <great circle> through these two points. Every critical path runs along it with constant speed, possibly passing around it extra times. If
$$
e=\frac{q-(\cos\theta)p}{\sin\theta},
$$
then a complete enumeration without repetitions is
$$
\gamma_\ell(t)=\cos(\ell t)p+\sin(\ell t)e,\qquad \ell=\theta+2\pi m,\quad m\in\mathbb Z.
$$
Its length is $L=|\ell|$. Equivalently the two positive length lists are
$$
L^+_r=\theta+2\pi r,\qquad L^-_r=2\pi-\theta+2\pi r,\qquad r=0,1,2,\ldots.
$$
There are no other critical paths, because a nonconstant <geodesic> of the round <sphere> is a constant-speed <great circle> and its plane must contain $p,q$.
To compute <conjugate points and indices of round-sphere geodesics>, the round unit <sphere> has constant <sectional curvature> $1$. Along a geodesic of speed $L$, a normal <Jacobi field> in a parallel direction satisfies $y''+L^2y=0$. A field with $J(0)=0$ is therefore a constant multiple of $\sin(Lt)$ in each of the $n-1$ normal directions. Its tangential component satisfies $y''=0$ and contributes no endpoint-vanishing field. Consequently the <conjugate points> occur at $Lt=j\pi$, each with multiplicity $n-1$. Since neither length list contains an integer multiple of $\pi$, all the displayed critical paths are nondegenerate, and the <Morse index theorem> gives
$$
\boxed{\operatorname{ind}(\gamma^+_r)=2r(n-1),\qquad\operatorname{ind}(\gamma^-_r)=(2r+1)(n-1).}
$$
The permitted case $p=q$ requires separate treatment. There is a constant geodesic, which has index and nullity zero. All other critical paths are
$$
\gamma_{r,v}(t)=\cos(2\pi rt)p+\sin(2\pi rt)v,
\qquad r\geq1,\quad v\in T_pS^n,\ |v|=1.
$$
For each $r$, this is a family parametrized by $S^{n-1}$; both directions of traversal are included through $v$ and $-v$. The interior <conjugate points> number $2r-1$, and the endpoint is also conjugate. Thus
$$
\boxed{\operatorname{ind}(\gamma_{r,v})=(2r-1)(n-1),\qquad\operatorname{nullity}(\gamma_{r,v})=n-1.}
$$
These are <Morse-Bott critical manifolds> for $n>1$, rather than nondegenerate critical points. For $n=1$ the same formulas give zero index and nullity, with the two isolated directions comprising $S^0$.
For the requested <homology> calculation, choose distinct nonantipodal endpoints. Concatenation with a fixed path back to the basepoint gives a <homotopy equivalence> between this fixed-endpoint <path space> and the based <loop space> $\Omega S^n$. We use the <Morse cell-attachment theorem for geodesic energy>: on a complete compact <Riemannian manifold>, the fixed-endpoint energy, when all its critical points are nondegenerate, gives a <CW complex> of the same homotopy type with one cell of dimension equal to the index of each critical point. One can obtain this theorem from finite-dimensional broken-geodesic approximations and ordinary <Morse theory>; critical energy values tend to infinity here.
The two index lists interleave to give exactly one cell in every dimension $j(n-1)$, $j\geq0$. If $n>2$, these dimensions are separated by at least two. The <cellular homology> groups therefore have zero boundary maps, since no occupied cell dimension has an occupied dimension one lower. With integer coefficients,
$$
\boxed{H_d(\Omega S^n;\mathbb Z)\cong
\begin{cases}
\mathbb Z,&d=j(n-1),\quad j=0,1,2,\ldots,\\
0,&\text{otherwise}.
\end{cases}}
$$
This determines the graded abelian groups. The argument does not need an identification of the multiplication on <loop-space homology>.
Finally apply the <Freudenthal suspension theorem>: for an $(n-1)$-connected based <CW complex> $X$, $n\geq2$, the suspension homomorphism
$$
\pi_k(X)\longrightarrow\pi_{k+1}(\Sigma X)
$$
is an isomorphism for $k\leq2n-2$ and a surjection for $k=2n-1$. A <sphere> $S^n$ is $(n-1)$-connected, and its reduced <suspension of a topological space> is homeomorphic to $S^{n+1}$. Hence
$$
\boxed{\pi_k(S^n)\cong\pi_{k+1}(S^{n+1})\quad(k\leq2n-2).}
$$
The isomorphism is the suspension map. Its stable range is also reflected by the <Morse theory> cell structure of $\Omega S^{n+1}$: after its bottom $n$-cell, the next positive-dimensional cell has dimension $2n$. For the stated homology problem $n>2$, all the connectivity hypotheses apply.
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