which WGS84 datum?

1. If you like. The different versions are not that different. Normally you lose more precision in the quantification of the coordinates then by using the wrong version of the datum. (And way more by using inaccurate measuring devices like a GPS.)
2. The API does not have an option of specifying a datum and any coordinates will be interpreted as WGS84. If a user uploads in the wrong datum it will not be converted. This is normally not a problem since any format the API supports (GPX, OSM) uses WGS84 and the error will therefore be in the converters.
3. The database stores GPX traces separately from vector data and all conversion between them is done by hand. The traces are a guide to the user when entering vector data and there are no automatic computation of vector data from the traces.

Please anyone feel free to correct me if I am wrong, but this is my understanding.

The WGS-84 datum is supposed to be centred at the earth's centre of mass and aligned with its axis of rotation. Satellite-based systems can determine those parameters to an accuracy of millimetres. It is also necessary to define, arbitrarily, the location of 0° of longitude (the Greenwich meridian). Since WGS-84 was defined (in 1984), the thing that has most affected its accuracy is tectonic plate movement (continental drift), which has altered the position of some major land masses by a significant fraction of a metre. That has resulted in several derived datums being specified, tied to significant land masses (North America, Europe, etc.). The re-specification of WGS-84 itself has, I think, resulted in changes of only fractions of a millimetre.

A good hand-held GPS receiver, operating under ideal conditions, can fix its position to within one or two metres. Under trees or near tall buildings, they are nowhere near that good. I have not heard of anyone using anything more accurate to add data to OSM, so it would appear that the consequences of the movement of the land relative to WGS-84 (away from major earthquakes) are about one order of magnitude smaller than the accuracy that can be expected from most OSM data, and the consequences of the re-specification of the WGS-84 datum are trivial. That is to say, we don't need to worry just yet.

As far as I know, positions in GPX files are always referred to WGS-84, and any device that writes a GPX file should convert locations to that datum, even if the device holds the information internally in some other form.