= Solution
Throughout the algebraic questions, $C$ is a smooth projective connected <algebraic curve> over the <algebraically closed field> $k$. This is the curve convention needed for the <Jacobian variety>; for an arbitrary singular or nonproper curve the assertions need not hold. Choose $P_0\in C(k)$ and write $\pi:C\times T\to T$.
The degree-$d$ <Picard functor of a curve> is
$$
\mathcal P^d(T)=\{\mathcal L\in\operatorname{Pic}(C\times T):\deg\mathcal L_t=d\text{ for every geometric }t\}/\pi^*\operatorname{Pic}(T),
$$
with pullback as its action on morphisms. Equivalently, normalize $\mathcal L$ by tensoring with $\pi^*(\mathcal L|_{P_0\times T})^{-1}$ and remember the resulting trivialization along $P_0\times T$. Since $\pi_*\mathcal O_{C\times T}=\mathcal O_T$, an automorphism of a <line bundle> is multiplication by a base unit; preservation of this trivialization forces that unit to be $1$. Thus normalized <line bundles> have effective descent and the displayed <functor> is already the appropriate sheaf, rather than just a list of classes over fields.
Here is a construction of its representing <scheme>. The <symmetric product of a curve> $C^{(r)}$ represents relative effective degree-$r$ <Cartier divisors>. It is projective, and its completed local ring at $\sum m_iQ_i$ is a power-series ring in the elementary symmetric coefficients of $m_i$ local parameters at each $Q_i$. Thus it is smooth of dimension $r$, in every characteristic. Put
$$
U=\{D\in C^{(g)}:H^1(C,\mathcal O_C(D))=0\}.
$$
This is open by the <semicontinuity theorem for coherent cohomology>. It is nonempty: choose $g$ points successively imposing independent vanishing conditions on $H^0(C,K_C)$, so that $H^0(C,K_C-D)=0$, and use <Serre duality>. The <Riemann-Roch theorem> gives $h^0(\mathcal O_C(D))=1$ on $U$.
For any fixed <line bundle> $B$ of degree $g-d$, use a copy $U_B$ of $U$ with the universal normalized <line bundle> $\mathcal O(D)\otimes B^{-1}$. This represents exactly the open subfunctor on which $H^1(\mathcal L_t\otimes B)=0$. Indeed, <cohomology and base change for line bundles on a curve> makes $\pi_*(\mathcal L\otimes B)$ a rank-one <vector bundle>. The evaluation morphism
$$
\pi^*\pi_*(\mathcal L\otimes B)\longrightarrow\mathcal L\otimes B
$$
has a unique relative zero <Cartier divisor> of degree $g$ and recovers the point of $U_B$. No choice of a global generator on $T$ is needed. Conversely that zero <Cartier divisor> recovers $\mathcal L$ modulo a bundle from $T$.
These charts cover: given any degree-$d$ <line bundle> $L$, choose a nonspecial effective degree-$g$ <Cartier divisor> $D$ and take $B=\mathcal O(D)\otimes L^{-1}$. On the overlap of $U_B$ and $U_{B'}$, the unique effective representative of $\mathcal O(D)\otimes B^{-1}\otimes B'$ is constructed by the same rank-one evaluation map. It therefore gives a regular transition morphism; uniqueness gives both the inverse and the cocycle identity. Gluing gives a <scheme> $J^d$ representing $\mathcal P^d$, with a universal normalized <line bundle>.
The construction is of finite type, not an infinite union of indispensable charts. Choose $m\ge2g-1$ and a fixed <line bundle> $F$ of degree $m-d$. Every $L\otimes F$ has a nonzero section by the <Riemann-Roch theorem>, so the universal effective <Cartier divisor> gives a surjection $C^{(m)}\to J^d$. The inverse images of the charts cover the projective, hence quasi-compact, <symmetric product of a curve> $C^{(m)}$. A finite subcover gives finitely many charts for $J^d$. This surjection also makes $J^d$ irreducible. The charts show that it is smooth of dimension $g$.
For properness, apply the <valuative criterion for properness>. Let $R$ be a discrete valuation $k$-algebra with fraction field $K$, and let $L_K$ be a degree-$d$ <line bundle> on $C_K$. A rational section represents it by a <Cartier divisor>. Its closure in the regular surface $C_R$ is again a <Cartier divisor>, since regular local rings are factorial. Thus $L_K$ extends to a <line bundle> on $C_R$, whose fibre degree remains $d$ by <constancy of line bundle degree in a family>. Two extensions with the same generic class differ by a vertical <Cartier divisor>. The special fibre is irreducible, so that <Cartier divisor> is an integer multiple of the entire fibre, the principal <Cartier divisor> of a power of a uniformizer. The extensions therefore give the same normalized Picard class. This proves existence and uniqueness in the valuative criterion, hence separatedness and properness of the finite-type <scheme> $J^d$.
Consequently \b[$\operatorname{Pic}^d_C=\operatorname{Jac}^d_C$ is a smooth proper $k$-variety of dimension $g$]. Tensoring by $\mathcal O_C(-dP_0)$ identifies it with $J^0$. On $J^0$ the <tensor product> and inverse of <line bundles> induce the algebraic group operations; it is the <Jacobian variety>, an <abelian variety>.
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