I have some questions about background subtraction with Gaussian Mixture Model (GMM)
Yes, the dimension of $X$ should be 3, one for each of the red, green, and blue channels. You can see this from their Figure 2, where they show plots of the distributions of red and green channels only (to motivate why they want to use a Gaussian Mixture Model in the first place --- the distribute of pixel values is multi-modal).
Yes, $I$ is the identity matrix.
Random variables can have the same variances and still be independent. The variance is part of the model. In this case what the model is saying is: the red, green, and blue pixel values vary without regard to each other (they are independent) but the amount that they vary is about the same as each other (they have the same variances).
Whether only 1 distribution is chosen when $K=3$ will depend on the threshold $T$, surely?  Yes, it is possible that only one distribution will be chosen.