By a quick search after reading your first paragraph:
In the rectangle from $(0,0)$ to $(56,63)$, the point $(20,48)$ is at integer distances from all the corners. Going clockwise from the origin, the distances are $52, 25, 39$, and $60$.
Edit. Corresponding Pythagorean triples (in the same order) are $20^2+48^2=52^2$, $20^2+15^2=25^2$, $36^2+15^2=39^2$, and $36^2+48^2=60^2$.
