Ah, the perils of houseplant ownership! Question: do you have other plants and do they show any signs of being affected the same as the pictured one.
There are many causes of what can cause brown leaf tips: from too much heat through the window glass, to too much, or too little watering. Too much fertilizer and any of the above.
You water on a schedule.....your schedule, not the plant's. Seasons change, the sun is going lower, the heat of our homes is being brought up when the furnace is brought on. They require less water as the season of growth diminishes....since the sun is less intense, the plant needs to get along with less.
I've never been a fan of artificial light, especially as it comes down from a ceiling having tube lighting--its just too far away to have any effect. Plants need light that is as close as inches....not feet. And it has to receive it from the centre of the spectrum.
About watering. When you water, try using your finger, down to the first knuckle at least....the second knuckle is being more kind, then water to drainage. Let the plant drain, then dump any excess in the saucer. Never leave the water sitting there for much longer than 10 or 15 minutes.
As far as the white specks, cant comment what it might be.....but pests usually cause damage that you should be able to see. Leaf tissue that turns a lighter green, or holes in leaves that show a pest is eating, or part of the leaves turn brown or yellow...maybe the lower leaves first since they are closer to the soil where eggs/larvae/adult stage of bugs come from.
Any sign of a stickiness on the leaves. That could be a sign of 'honeydew'...secretion from the pest and is a danger if not controlled.
Since there is many pests and many remedies from them, it can be dangerous to spray for one, and not find the right one, its sometimes better to let the bug have its bite, then move on than to spray something that is dangerous to the whole plant.
The one method you can safely use for all, is WATER. Put some water into a hand sprayer and hit the plant hard. Knock off the bugs, then find out what bug you have to best treat. The brown of the tips should be scissored to keep the dead material from inviting further damage. If it is a pest, isolate it, so to keep your other plants safe from cross infestation.