I'm not sure about the coffee rust (I'm new to coffee, but maybe a fungicide?)
The insects sound like mealybugs. Pain in the butt! But treatable. :) The first general product I'd recommend to kill them is neem oil. You can buy it pre-mixed, or you can get concentrate and mix it up in a spray bottle (Cheaper and lasts longer). Spray the whole plant in the shower - EVERY inch of it. Soak it. Let it sit overnight, and then rinse in the morning. You'll have to do this once a week for a few weeks to get any that might have been missed.
If that doesn't work, you can try horticultural oil. But the big guns are these systemic insect control granules that actually treat the plant from the inside and make the plant toxic for the mealybugs to consume, eradicating them. The only caveat is that you should keep the plant away from kids or animals that might chew the leaves while it's active.