A dimensionless number has no units. The units of all variables that compose the dimensionless number (product or ratio) must cancel each other.
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Correct: it is a dimensionless number.
Yes. Conversion factors will generally be dimensionless constants.
infinity
No, but an angle is dimensionless.
Specific heat is dimensionless, and dimensionless units have the same value in any system. Specific heat is the ratio between two densities - that of the substance considered, and that of water. The ratio of two quantities of the same dimension will naturally be a dimensionless number.