K-means Clustering algorithm problems
Wikipedia says: "Assign each observation to the cluster whose mean yields the least within-cluster sum of squares (WCSS)"

I think in your case, this is translatable to: $c_i$ is assigned to the closest centroid by euclidean distance.

For your second question, the centroid should $\mu$ should have the same number of dimensions as each training point $x_i$. They are both points in the co-ordinate system.

You can use a high number of features with K-means, for example, text analytics might reduce a corpora of news articles to 10,000+ dimensions. Depending on the package you use these might be represented as a sparse matrix.