I'm doing the HQ mission, and I've got member files, and now want to get member medical files. I go to the room where they are stored, I attempt to pick the lock, but it seems to do it automatically (I don't pick the lock, it seems like he is doing it himself), then when that is finished, I stand there still, and I can't move at all. The timer is still going, and I can load from last save which was luckily moments before this, but I want to get the records, the game isn't letting me.
I've not had the 'hangup' as you describe it, but once you have completed certain 'star' objectives you get new equipment unlocked. One of those is the 'automatic lock pick', so if you check your unlocked equipment list on your OPSAT, you'll probably see that you've acquired it.
As to the game 'hanging' there, you might be able to wait for HQ 4 to see if you can get it there. I usually acquire them on HQ 1 rather than wait.