How to compare the performance of different number of mixing components for EM algorithm?
1.

The simplest and most common way is to use AIC or BIC. You would pick a model with the minimum AIC/BIC value. AIC/BIC work well here because you have a likelihood function.

Bayesian model selection is another possibility. It's more advances than AIC or BIC, but you get chance to add your own prior distribution. Section 5.3 in 


  Machine Learning - A Probabilistic perspective (Kevin P.Murphy)


has the details. There're also lots of papers on Google.

Alternatively, you can use the cross-validated likelihood as a performance measure, although this can be slow, since it requires fitting each model N times, where N is the number of CV holds.

Slide 17 in


  https://www.doc.ic.ac.uk/~dfg/ProbabilisticInference/IDAPISlides13.pdf


is a good reference.

2.

EM algorithm is a technique for estimating the most likely parameters and is not limited to mixed Gaussian. The algorithm is useful when there's no closed form formula for the solving maximum likelihood problem with one or more latent variables.