This is an update on Question 1. As Willie observed, in his 1819 memoir Poisson studies not only the wave equation but also the heat equation from page 143 on, and reaches the heat kernel on page 145. However, amazingly, in Fourier's original memoir where he derived the heat equation and gave a convincing case for the importance of trigonometric series, the heat kernel appears on page 454 for 1D, on page 475 for 1D in the usual form as presented today, and on page 479 for 3D. Fourier's memoir was published in 1822 after a long delay, and it is said that the memoir is essentially Fourier's 1811 work that won a mathematical prize, which was in turn a continuation of his work presented in 1807, and summarized by Poisson in 1808. That said, even more amazingly, a new player appears in the story. After giving the 1D heat kernel on page 454, Fourier says something like
This integral, which contains an arbitrary function, was not known when we started our research on the theory of heat, which were presented at the Institute of France in December 1807. It was given by Mr. Laplace, in volume VI of des Mémoires de l'école polytechnique, and we have only applied his results here.
Poisson also mentions Laplace on page 148, and says that his 3D result was a straightforward extension of Laplace's formula. I found volume 6 of Journal de l'école polytechnique (not des Mémoires) but there is nothing by Laplace, and moreover the journals journal is from 1806. I also do not know wondered if Journal and des Mémoires are is different than Journal, but skimmed through Fourier's book to find that on page 513 he cites Laplace again, but now says volume 8. Then volume 8 it is! It is published in 1809, and the heat kernel appears on its page 241!