Is It Possible to Reconstruct the Original Image from Laplacian Pyramid?
This depends on the order of upsampling and downsampling.
If the order is correct, then you won't throw away anything and thus you should in principle be able to reconstruct the image.

In general:

$$ \left(\uparrow_n\downarrow_n f\right) \neq \left(\downarrow_n\uparrow_n f\right) $$

Similar things are used when using the Wavelet decomposition on a signal, which again uses up- and downsampling combined with the Wavelet filters.