No, for example a 36MP FF and 18MP FF using the same manufacturing technology, the 18MP FF will have better low-light properties as which pixel received more light than the 36MP FF.
Not true either - the world is awash with examples that prove the opposite: Nikon's D7000 has clearly superior how light performance to the D300; the Canon 70D is much better than the 30D; the 1D Mk IV is far superior to the 1D II!n.
And so on.
Smaller pixels do not mean inferior low noise performance.
Even DxO gets it:
To draw conclusions about pixel size and noise you must compare sensors with the same technology. We know, for example, current Nikon sensors have much better S/N than equivalent Canon at low ISO. So comparing Nikon with Canon is misleading.
The DXO article says that the Canon 350D and 1Ds have identical sized pixels (6.4 micron) and identical S/N, which could be interpreted as it is pixel size that determines S/N. It then goes on to say that at the same field of view the 1Ds has better S/N than the 350D when both images are printed at the same size. That is not due to the pixel size but results from the larger sensor of the 1Ds.