The shifter shakes because it's bolted directly into the transmission. So, as the engine and transmission vibrate, so does the shifter. If it bothers you, harder engine mounts will square this up, but they'll make the rest of the car vibrate more in turn.
S13? S14? If it's not an '91-'93 S13 SE/LE with SuperHICAS and ABS, or an S14 SE/LE with ABS, then you have an open diff, with 99% certainty.
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