So when I play Divide quest, I saw the notification pop up saying I got a buff based on the maximum number of players in the area. Sometimes, I check my stats it has the buff, but sometimes it doesn't have. What's happening? Is it a one-time buff per area, or reset per life, or reset on swapping weapon?

it's a one time buff per area that activates like about after a minute and a half into the stage.

Apparently, in some stages, you will need to kill small mobs to gain attack buff.

Other than the final step that is every multiple of 5, you get the attack buff 1min and 30seconds in as it is technically still matching time and random people can join in still.

There's a PSE Atk Buff you get in Boss and Infiltration Stages for killing the small mobs / activating virtual devices.

That one is shown in your ATK stats. The player count buff comes 1:30 after the stage starts or instantly at Final stages, but they don't show in your stats