Do I understand right that the verb jít (here jde) does not imply a direction? In English, go and come convey some sense of direction. Please compare:
He is going to the supermarket.
He is coming from the supermarket.
He is going from the supermarket.
He is coming to the supermarket.
In each instance, there is some indication of where he is based on the combination of verb and from/to. The last two are less common formulations: going from means he is leaving, and coming to means he is arriving.
I have a never-ending struggle with my German-speaking wife concerning bring and take, which seem to work exactly the opposite of bringen and nehmen in German. :)
I just want to make sure I understand that jít simply means to go by foot and jet means to go by vehicle, without implying any sense of direction. In other words, we only know that "Žofie is coming here" and not going somewhere else because of the word sem.
My Polish is far from great but I consider “tu” and “tutaj” largely synonyms, meaning the place where the speaker is; “tu” seems to be used also in a directional sense, meaning “to the place where the speaker is.” You probably know better.
In Czech, “tady” is stationary (in this place) and “sem” is directional (to this place). That's all I can say.
Firstly, you probably mean TADY, not LADY.
Secondly, tady is a location (here, at this place) while sem is a direction (hither, to this place). You cannot mix them (although some dialects may do that). - See the rest of the discussion for this point, it has already been discussed.