EDIT - Disclaimer: I would like to refer the readers to the discussion with ChrisW below. It might be that getting an area based upon an OTF CRS is not a bug after all; that is, at least, in arcgis it also being used to allow geoprocessing two layers from different CRS.
To elaborate on the issue above. As AndreJ as suggested and show - this is probably a bug in qgis's current version. Yet it should be noted that the problem isn't the wrong area, but that on-the-fly transformation affects in anyway on area calculations.
The purpose of on-the-fly transformation/projection is to align data from different sources and with different CRS. That is mainly for display purposes. E.G. arcmap automatically perform on-the-fly projection in any case a layer CRS does not match the data frame CRS.
Arcmap also provides a possibility to edit data while projected on-the-fly, but also notes that: (source)
However, it is important to note that certain editing operations may produce unexpected alignment or accuracy problems, depending on the coordinate systems being used.
Specific editing operations that may cause issues include changing the shapes of features, snapping to the edge or boundary of features, or extending and trimming features. These problems are more likely to occur when the features you are editing are close to the edge or beyond the area of use of the coordinate system
That is to say: on the fly transformation is less acurate than just project the data to a different CRS (which also introduces its own problems).
Having said that it is not surprising that based on a on-the-fly transformation a wrong area is being calculated, yet it is surprising that the fact that on-the-fly was enabled affects in any way the calculation of geometry, which should be based on the data. Thus, it doesn't matter if on-the-fly transformation is based upon the same or a different CRS, area calculation should be identical each time.
To be more practical, if your aim is to compute the area do not use on-the-fly. If you have the wrong CRS, project your data.
Is it about the calculated area or the view?
The calculated area should be nearly the same (on smaller extents up to a few hundred kilometers) as both are using nearly the same geoid (minimal difference between GRS 80 and WGS 84) for calculation. What you see on the screen is a matter of projection and does not have anything to do with the real area. When you try to get the same "view" in different projections (or in your case no projection vs mercator) you are creating entirely different polygons.
Best Answer
If the values are first very big and then negative, you are most likely running into an integer overflow problem. Are the polygons very big in reality? Maybe you shouldn't calculate square meters but at least kilometers.
for km²:
$area / 1000000for ha:
$area / 10000On-the-fly reprojection is not needed for this task.