Yup, you have spider mites.
The mold you refer to is actually calcium deposits. Due to hard water and/or the pot standing in water the soil evaporates water and leaving scale to form on the surface. In itself not too big of a problem. It can over time lead to problems like inability of the tree to take up nutrients.
Do you have very hard water? Or is your pot sitting in water? Make sure it does not.
How do you water? Do you water every day a little? With the coarse subatret you will have to water often, but also generously. Which is harder to do keeping it inside (By watering more than the pot can hold, you help flush out part of the salts in the pot, reducing the buildup).
The spider mite.. They are tough. You will need to spray it probably 3 times at the right interval to get rid of the infestation. And you might find you get them again soon after. Make sure you also clean the outside of the pot and the areas where the pot sits when you spray to get any of them lurking on the windowsill. They are a sign of dry windless conditions, which match your indoors placement of the tree. DO you have the option to keep it outdoors? (Depending on what winters you get there though!)