It seems that Euler has completely answered your first question: see the [slides][1] to a talk given by Noam Elkies, who credits Franz Lemmermeyer (presumably for bringing this under his attention). 

The surface described by Chris Wuthrich has a K3 surface as its smooth model. Euler uses the presence of rational curves on it to produce infinitely many solutions (as suggested by Will Sawin).

[In the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, the equation becomes the more symmetric $$
pqr(p+q+r) = A^2.
$$
I knew I recognized this equation from somewhere, and after a little searching I hit upon Noam's presentation.]


  [1]: http://www.math.harvard.edu/~elkies/euler_14t.pdf