Partitioning a Rectangle into Congruent Isosceles Triangles

No. Note that the acute angle of your triangle must divide $\pi/2$ (look at a corner), so there are countably many such triangles (up to similarity), and hence you get only a countable set of possible ratios of sides.

If the length divided by the width is rational, then yes. Just partition the rectangle into congruent squares and cut each square along a diagonal.