Not “though space”, there is no inherent stationary frame of reference. Solar system is moving around mass center of Milky Way at about 220km/s and yes, so does everything in solar system, everything from rockets to teddy bears included.
If however we don’t care about center of Milky Way and only look at Earth’s orbit around Sun, then that speed is only 30km/s
Speed is always relative to something else. It’s just confusing because in everyday vernacular we are used to assuming that speed is relative to Earth, making us treat speed almost as if it were an absolute quantity, it isn’t.
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