Can gaining core strength and stability increase speed?and....

What does it mean when you look stiff when you run?

Increasing core strength (low back, obliques, abdominals, etc) will definitely aid in enhancing the force/posture posture relationship, however, as far as increasing an athletes speed goes; this would be relative to the abilities of the subject in question.

‘Looking stiff’ is most likely a lack of dynamic flexibility/technical efficiency.

Be like water…