  Stellar variability impacts radial velocities at various timescales and
therefore the detectability of exoplanets and the mass determination based on
this technique. It is necessary to implement systematic studies, to delineate
the current limitations of RV techniques to detect Earth-like planets. This
paper aims are to investigate whether the targeted 10% mass uncertainty from RV
follow-up of transits detected by PLATO can be reached, and to analyse and
quantify Earth-like planet detectability for various spectral types. We
implemented blind tests based on a large data set of realistic synthetic time
series reproducing different phenomena leading to stellar variability such as
complex magnetic activity patterns as well as flows, covering F6-K4 stars and a
wide range of activity levels. The 10% mass uncertainty for a 1 MEarth in the
habitable zone of a G2 star cannot be reached, even with an improved version of
a usual correction of stellar activity and even for long-duration (ten years)
well-sampled observations. This level can be reached for masses above 3 MEarth
or for K4 stars alone. We quantify the maximum dispersion of the RV residuals
needed to reach this 10% level, assuming the correction method and models do
not affect the planetary signal. Several other methods were tested and do not
allow a significantly improvement of this limited performance. Similarly, such
low-mass planets in the habitable zone cannot be detected with a similar
correction: blind tests lead to very low detection rates for 1 MEarth and a
very high level of false positives. Very significant and new improvements with
respect to methods based on activity indicators to correct for stellar activity
must be devised at all timescales to reach the objective of 10% uncertainty on
the mass or to detect such planets in RV. Methods based on the correlation with
activity indicators are unlikely to be sufficient.
