Stars rotate due to the angular momentum of the gas they formed from. This angular momentum must be conserved, and remains as the rotation of the star and it's satellites. If a star collapsed from a completely static gas cloud it would not rotate, but would still be stable. The stability is provided by the hydrostatic equilibrium between radiation and thermal pressure with gravitation collapsing the star - not the stars rotation.
I don't know any statistics but I expect all stars rotate to some degree; It is usually the turbulence of gas clouds that leads to overdensities that then collapse in to stars.
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