= Solution
State the characteristic convention explicitly: for real $\alpha,\beta$, define the <theta function with characteristics> by
$$
\boxed{\theta[\alpha,\beta](z,\tau)=\sum_{n\in\mathbb Z}\exp\bigl(\pi i(n+\alpha)^2\tau+2\pi i(n+\alpha)(z+\beta)\bigr),\qquad\operatorname{Im}\tau>0.}
$$
Binary subscripts $\vartheta_{ab}$ often denote this function with $(\alpha,\beta)=(a/2,b/2)$, $a,b\in\{0,1\}$. On compact subsets of the $(z,\tau)$ domain, a summand and any derivative are bounded by a polynomial in $|n|$ times $e^{-c n^2+C|n|}$. Normal convergence therefore gives an entire function of $z$, holomorphic dependence on $\tau$, and legitimate termwise differentiation. In particular the <heat equation for theta functions with characteristics> is
$$
\boxed{4\pi i\,\partial_\tau\theta=\partial_z^2\theta.}
$$
Characteristics are defined modulo integers with a phase: $\theta[\alpha+1,\beta]=\theta[\alpha,\beta]$ and $\theta[\alpha,\beta+1]=e^{2\pi i\alpha}\theta[\alpha,\beta]$. Shifting the summation index proves the spatial quasi-periods
$$
\theta(z+1)=e^{2\pi i\alpha}\theta(z),\qquad
\theta(z+\tau)=e^{-\pi i\tau-2\pi i(z+\beta)}\theta(z).
$$
These multipliers describe a degree-one <line bundle> on the <complex torus> with periods $1,\tau$, rather than an ordinary elliptic function. The logarithmic derivative is unchanged under one and decreases by $2\pi i$ under tau. Pairing opposite sides of a fundamental cell therefore makes its contour integral $2\pi i$. The <argument principle> counts exactly one zero in a cell, with multiplicity.
The characteristic function is a translate of the unshifted theta function:
$$
\theta[\alpha,\beta](z,\tau)=e^{\pi i\alpha^2\tau+2\pi i\alpha(z+\beta)}\theta[0,0](z+\alpha\tau+\beta,\tau).
$$
At $z=(1+\tau)/2$, the unshifted series cancels in pairs $n,-n-1$. Hence the <zeros of theta functions with characteristics> occur exactly and simply at
$$
\boxed{z=(1/2-\alpha)\tau+(1/2-\beta)\pmod{\mathbb Z+\mathbb Z\tau}.}
$$
Reindexing also gives $\theta[\alpha,\beta](-z)=\theta[-\alpha,-\beta](z)$. For half-integer characteristics the parity is $e^{4\pi i\alpha\beta}$, so only $(1/2,1/2)$ is odd and vanishes at zero; the other three theta constants are nonzero.
The <modular transformations of theta characteristics> are obtained from Gaussian <Poisson summation> and elementary phase rearrangement. With the holomorphic branch of $\sqrt{-i\tau}$ positive at $\tau=i$,
$$
\boxed{\theta[\alpha,\beta](z/\tau,-1/\tau)=\sqrt{-i\tau}\,e^{\pi iz^2/\tau+2\pi i\alpha\beta}\theta[\beta,-\alpha](z,\tau),}
$$
$$
\boxed{\theta[\alpha,\beta](z,\tau+1)=e^{-\pi i\alpha(\alpha-1)}\theta[\alpha,\beta+\alpha-1/2](z,\tau).}
$$
For the first formula, apply Poisson summation to $F(t)=e^{\pi i\tau(t+\alpha)^2+2\pi i(t+\alpha)(z+\beta)}$. Completing the square in its Fourier transform gives
$$
\widehat F(m)=(-i\tau)^{-1/2}e^{2\pi im\alpha-\pi i(z+\beta-m)^2/\tau}.
$$
Rearranging $\sum_nF(n)=\sum_m\widehat F(m)$ proves the stated inversion law. For translation by one, use $e^{\pi in^2}=e^{\pi in}$ in the defining series; the remaining characteristic phase is precisely the displayed factor.
Write $Q=e^{\pi i\tau}$, so the modular-form variable of Question 1 is $q=Q^2$, and put $\Theta_2=\theta[1/2,0](0,\tau)$, $\Theta_3=\theta[0,0](0,\tau)$ and $\Theta_4=\theta[0,1/2](0,\tau)$. Inversion interchanges $\Theta_2,\Theta_4$ and fixes $\Theta_3$, each with factor $\sqrt{-i\tau}$. Translation interchanges $\Theta_3,\Theta_4$ and multiplies $\Theta_2$ by $e^{\pi i/4}$. The <Jacobi triple product> gives
$$
\Theta_2=2Q^{1/4}\prod_{m\geq1}(1-Q^{2m})(1+Q^{2m})^2,
$$
$$
\Theta_3=\prod_{m\geq1}(1-Q^{2m})(1+Q^{2m-1})^2,\qquad
\Theta_4=\prod_{m\geq1}(1-Q^{2m})(1-Q^{2m-1})^2.
$$
It also gives the conventional odd theta function $\theta_1=-\theta[1/2,1/2]$ its product $2Q^{1/4}\sin(\pi z)\prod_{m\geq1}(1-Q^{2m})(1-2Q^{2m}\cos(2\pi z)+Q^{4m})$. Differentiate at zero and multiply the three even products to obtain the <Jacobi derivative formula>
$$
\boxed{\theta_1'(0,\tau)=\pi\Theta_2\Theta_3\Theta_4.}
$$
In the constant-product simplification, $\prod_m(1+Q^{2m})(1-Q^{4m-2})=1$, by separating even and odd positive exponents. The products prove nonvanishing of all three even constants as well.
The <Jacobi abstruse identity> can be proved directly, without assuming it from the products. Splitting pairs of summation indices according to whether their sum and difference are even or odd gives
$$
\Theta_3(\tau)^2=\Theta_3(2\tau)^2+\Theta_2(2\tau)^2,\qquad
\Theta_4(\tau)^2=\Theta_3(2\tau)^2-\Theta_2(2\tau)^2,
$$
$$
\Theta_2(\tau)^2=2\Theta_2(2\tau)\Theta_3(2\tau).
$$
For example, $u=(n+m)/2,v=(n-m)/2$ are either both integers or both half-integers, and $n^2+m^2=2u^2+2v^2$. The alternating signs give the second identity; shifting both original indices by one half gives the third. Squaring the first two and subtracting then proves \b[$\Theta_3^4=\Theta_2^4+\Theta_4^4$].
These identities connect theta theory to <modular forms> and <elliptic curves>. The fourth powers are weight-two forms on the principal level-two <congruence subgroup>. The <modular lambda function> $\lambda=\Theta_2^4/\Theta_3^4$ obeys $\lambda(-1/\tau)=1-\lambda(\tau)$ and $\lambda(\tau+1)=\lambda(\tau)/(\lambda(\tau)-1)$; the characteristic permutation kernel is $\Gamma(2)$. It parametrizes an elliptic curve with its ordered two-torsion in Legendre form $y^2=x(x-1)(x-\lambda)$. The corresponding <Klein j-invariant> is $256(1-\lambda+\lambda^2)^3/[\lambda^2(1-\lambda)^2]$. Finally the same products give
$$
\boxed{\Delta(\tau)=2^{-8}(\Theta_2\Theta_3\Theta_4)^8.}
$$
Indeed $\Theta_2\Theta_3\Theta_4=2Q^{1/4}\prod_m(1-Q^{2m})^3$, whose eighth power divided by $256$ is $q\prod_m(1-q^m)^{24}$. This recovers the normalized modular discriminant of Question 1 and displays the bridge between Gaussian theta sums and the lattice discriminant of Question 2.
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