In order to leave the lab, you are supposed to receive a keycode from Daedalus after finding Paul. The keypad to leave the facility can potentially be hacked, which makes this potentially an optional objective. Depending on whether or not you saved Paul, you will either be looking for (living) Paul or his (dead) body. You may also want to find your old equipment, which will be either in the Armory or the Lab (this isn't random, it just always spawns away from Paul).

Paul can sometimes be found on the upper floor of the Armoury to the South. If he is here then your equipment will be in the Lab.

Paul can sometimes be found in his standard location in the lab to the North. If he is here then your equipment will be in the Armory.

Paul can sometimes be found in the watery pit through the vents below the Nanotech lab. If he is here then your equipment will be in the Armory.

Paul can sometimes be found in the offices above the Robot Maintenance facility. If he is here then your equipment will be in the Lab.

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In order to leave UNATCO HQ, you need to get a nanokey from Alex Jacobson and as such, his location has been randomized. Jaime's location has also been randomized into the same pool, you need to talk to him if you want to get his gifts later in the game.

Alex can appear between the jail cells

Alex can sometimes be found hiding in the men's bathroom of UNATCO HQ

Sometimes Alex will be found in his office, which is where he is normally found

Alex can sometimes be found grabbing a drink in the break room

Sometimes Alex needs a bit of privacy and likes to use Manderley's private bathroom

Alex will occasionally hang out in the office on the west side of the upper floor

Anna's killphrase is 2 different parts, in 2 separate computers. Note that you will need to use the DEMIURGE username to access this. There is a datacube in the MJ12 Lab with the password, or Paul will give you the password if Paul and Anna are both still alive. On Manderley's desk there is a datacube saying which computers the killphrase parts are stored in.

Your door is locked, hopefully you got the "Office Door Key" back in Mission 1 UNATCO HQ!