According to reports on flying forums checking is being processed manually with huge queues as a result, flight status screens aren't working, etc.

Until an hour or so, ago login to the website was down for me.

So it seems it's a very widespread outage that they're in the process of recovering from

Likely they turned both of those off on purpose. If you can't depart due to lack of a manifest, you turn off flight status and check-in.

I was able to do a flight status a little while ago though.

If that's true then it's probably their operations system, i.e. flight management, digital flight bag, etc. Without those systems you can't file flight plans easily, fuel the planes with the optimal amount for the weather conditions, etc.

I would guess some component of what they call "Departure Control".