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"In my language it should mean царствовать (to be, to exist). Do you agree?"

You can say so in russian too, but it will be very poetic or exaggerated. "Стоять" about weather is very usual thing...

(note: I put ' before stressed letter below)

In these examples "сто'ит" is form of "сто'ять" verb (to stand). It's meaning in these cases is similar to "to reside / to exist / to stay...