Question 2 is getting clearer now. My sources are Parseval's article from 1800, Poisson's memoire from 1819, Hadamard's Lectures on Cauchy's Problem in Linear Partial Differential Equations (1923), and Baker and Copson's The Mathematical Theory of Huygens' Principle (1939).

On page 133 of the afore-mentioned memoire, Poisson gives the 3-dimensional formula

$$
u(x,t) = t M_{x,t}u_0 + \partial_t (t M_{x,t}u_1); \qquad u_0(x) := u(x,0), \quad u_1(x) := \partial_tu(x,0),
$$

where $M_{x,t}g$ is the average of $g$ (defined in $\mathbb{R}^3$) over the sphere centred at $x$ of radius $t$. Then he goes on to prove it, and by the method of descent, derives several special cases, including the 1 and 2 dimensional formulas. So the *3D case is due to Poisson*.

Later in 1882, Kirchhoff published a more general formula expressing $u(x,t)$ in terms of the values, the normal and time derivatives of $u$ over an arbitrary closed surface containing $x$, therefore mathematically justifying the Huygens principle. The analogue of Kirchhoff's 1882 formula for 2 dimensions was published by Volterra in 1894. These developments were closely related to the discoveries of fundamental solutions of the Helmholtz equation in 3 dimensions by Helmholtz in 1859, and for 2 dimensions by Weber in 1869.

As for who was the first to discover the 2 dimensional analogue of Poisson's 1819 formula, when he coins the term "method of descent", Hadamard notes

Creating a phrase for an idea which is merely childish and has been used since the very first steps of the theory is, I must confess, rather ambitious;

and cites Parseval's afore-mentioned article of 1800, Poisson's memoir of 1819, and Duhem's book from 1891. After giving the 2D formula on page 141 of his memoir, Poisson cites Parseval's article, and says something like "Parseval previously integrated this equation but in a less simple way". Parseval seems to give the formula on page 519 of his article, but I don't understand sufficiently to say the formula is complete. In particular there seem to be no explicit formulas for the quantities Q and Q'. So the *2D case can be said due to Parseval-Poisson*.