Hi Sarmed,
The first problem can be corrected by combining the styles using Directive:
OnOffOffPlot =
LangPlot =
ErrorListLogLinearPlot[{OnOffOff}, PlotLegends -> {"OnOffOff"},
PlotRange -> Full, Frame -> True,
FrameLabel -> {"Frequenz [Hz]", "Amplitude [mV]"},
LabelStyle -> {FontFamily -> "Times New Roman", 12, GrayLevel[0]},
ImageSize -> Medium,
PlotStyle -> Directive[Green, Thickness[0.002], PointSize[0.007]],
Joined -> True, GridLines -> Automatic,
GridLinesStyle ->
Directive[Dotted, GrayLevel[0.80], Thickness[0.002]]]

The second problem can be corrected by specifying the styles for the two data sets seperately as two lists within a list.
OffOffOffPlot =
ErrorListLogLinearPlot[{OnOffOff, OffOffOff},
PlotLegends -> {"OnOffOff", "OffOffOff"}, PlotRange -> Full,
Frame -> True, FrameLabel -> {"Frequenz [Hz]", "Amplitude [mV]"},
LabelStyle -> {FontFamily -> "Times New Roman", 12, GrayLevel[0]},
ImageSize -> Medium,
PlotStyle -> {{Thickness[0.002],
PointSize[0.007]}, {Thickness[0.002], PointSize[0.007]}},
Joined -> True, GridLines -> Automatic,
GridLinesStyle ->
Directive[Dotted, GrayLevel[0.80], Thickness[0.002]]]

Kind regards,
David