How can I improve my downhill mountain bike as a commuter?
You really can’t change it that much. A dual-suspension downhill mtb will always have limitations, most notably its weight, wheel size and width,  and geometry. 

Instead, get a used, 1970s, steel, road bike for your commute. It should cost you roughly $50-150 euros or dollars. 

You’ll be amazed at how light a road bike (even a steel antique) will feel compared to a mountain bike. My 1975 Peugeot MX-18 weighs just under 11kg (I did swap the steel rims for alu ones). This is half the weight of your mountain bike!

A road bike, even an old one, will be faster over paved surfaces and learning how to “float” over cobblestones and rough surfaces will help your mountain biking skills. 

Finally, since a used beater is so cheap, you won’t worry as much riding it in all weather conditions or about bike thievery.