Whether convolution followed by down sampling equals wavelet decomposition
Simply put, no!


  I know that wavelet decomposition equals convolution followed by down sampling by two


No again. Some linear wavelet decompositions can be turned into a cascade of convolutions followed by downsampling. But wavelets are much more general. Their discrete implementations involve intricate properties in filters, downsampling, and levels.

If you just filter a signal  by some filter, and downsample, in general, you get almost nothing but aliasing.