MsDoe said: Welcome Chef!
The white "mold" looks to me more like mealy bugs. I'd start by taking a q-tip dipped in rubbing (isopropyl) alcohol and wiping them off. Check the plant every few days and repeat as necessary.
In my experience, a ceramic pot with a single drainage hole tends to keep the soil too wet for succulents. You need to be very careful with watering. I'd recommend backing off to every two weeks and see how it does. In cooler weather it might need even less. Water thoroughly when you do water, then let the soil get quite dry before watering again.
Aloe vera grow well in bright outdoor shade/part sun. Indoors, it's very difficult to get it enough light to stay healthy, even by a window. I'd put that grow light on a timer and give it at least 12 hours a day, regardless of sun or clouds outside. Ramp this up slowly, so the plant can adjust.
If you decide to put it outdoors, put it in a shady spot and gradually acclimatize it to the increased light levels.
Be sure to get it back indoors when temps get below about 55 F.
Welcome to the site, and good luck with the Aloe!