When the pick a door game first popped up I clicked the door I wanted and waited for the next one. I did get a message saying, from the first door, that I could carry on, so I went to pick the next door (when it was open). Only whenever I clicked the next door in the 'second' round, nothing happened. It doesn't let me pick the first door, doesn't let me pick the second, nothing at all happens when I click a door. I suspect it's a bug because I tried clicking on the second door in another phase, but I don't know why it won't just let me choose the first door again to start over.

Not really a huge deal, but I am confused that I can't click ANYTHING on that gamescreen.

Any ideas?

Is it possible that in the first game you played, you waited too long to pick your next door and it had moved onto the next door set by that point?
Pick a Door is timed, so, if you miss the next round you were supposed to participate in, you won't be able to pick a door since it would have moved on to the next round/door set by that point.
The first set of doors you pick from has a longer waiting time between it and the next set of doors picked from to allow more people to participate in a round. This being three hours.
The second set gives you twenty minutes to pick a door.
Then sets 3-5 give you ten minutes to pick a door.
You'll still get a prize(If I remember right, you'd get a prize from the round you missed out picking a door on).

In particular, these were the listed schedules for today(Assuming the rounds you participated in were from today):

Game 9 Schedule:
Round 1: From 9:00 AM until 12:00 PM
Round 2: From 12:00 PM until 12:20 PM
Round 3: From 12:20 PM until 12:30 PM
Round 4: From 12:30 PM until 12:40 PM
Round 5: From 12:40 PM until 12:50 PM

Game 10 Schedule:
Round 1: From 3:00 PM until 6:00 PM
Round 2: From 6:00 PM until 6:20 PM
Round 3: From 6:20 PM until 6:30 PM
Round 4: From 6:30 PM until 6:40 PM
Round 5: From 6:40 PM until 6:50 PM

Game 11 Schedule:
Round 1: From 9:00 PM until 12:00 AM
Round 2: From 12:00 AM until 12:20 AM
Round 3: From 12:20 AM until 12:30 AM
Round 4: From 12:30 AM until 12:40 AM
Round 5: From 12:40 AM until 12:50 AM

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You can also only play one pick-a-door game per series, so trying to participate in another won't work, even if you missed your carry on in a previous round.

~

I hope this helps, though if after explaining it still seems like you experienced a bug, I very much recommend posting in the Bug Reporting forum to see if your issue can be resolved/explained:
https://www.sylestia.com/forums/?forum=8

Yeah, I believe I did that, where I chose the first door in the one round, that finished, then I tried to choose the second door in a whole other round, so I'll try again while closely watching the time for these. Thanks!