Relationship between order of filter and amplitude/phase response
Does a higher order simply mean a better approximation to the optimal filter with (cutoff, stopband) ? 


That depends on how you define "optimal". In general a high order will give you a steeper rolloff


  Do higher orders mean different or less severe phase differences?


Higher order will result in more phase distortion


  Finally, I'd assume that it affects performance somewhat as well.


That depends on how you define "performance". The "best" filter always depends on the application. 

In general higher IIR filter order will give you more steepness in the frequency but also more time smear and phase distortion. Any beginner class on digital filter should cover this. For example, take a look at https://ccrma.stanford.edu/~jos/filters/