Re: When entering a room.
At least from a veteran police officer's perspective, scanning a room to look for things out of the ordinary, prospective dangers, and general layout was my practice. I specifically did not make eye contact with the inhabitants unless I perceived a challenge; physical, oral, or visual. After the scan revealed no obvious dangers, I would make eye contact with the bartenders, and wait staff.
The down side is that it is very hard to turn off the practice. I got "burned" my first time undercover when I scanned the bar as I entered. The bartender recognized the cop behavior and loudly asked "What can I get you officer?"