Right now in game it's winter, around late Feb. 
Time of day is somewhat irrelevant because we do have several stories going on all at once, which in theory could take place at different times during the day. However, for the most part we all seem to be involved in stories at night.
Key things going on at the moment: Nightmare, Archangel, and Iron Knight just had a major brawl, The Cowl has sent is men out to raise some hell, and Lyger just gave some such men a message to deliver to their boss.
There's an epic battle of the gods brewing, an indescribable cosmic evil heading our way, and halfway across the world, the nation of Charonia has descended into civil war. 

And yes, Sherman Square, Lost Haven's own version of Times Square does seem to have issues with superpowered brawls taking place quite frequently...or at the very least something always seems to get blown up in the area...maybe that's another reason you got your apartment so cheap.