Some odd behavior

Nope, not normal. I've had mine for about 2 years and haven't had a single problem. Just keep an eye on it. Sometimes electronics are weird and just need to be reset from time to time.

Did you check your Timeline of Events to find out which sensor tripped the alarm?  Start there.  Then check your siren - try re-seating the batteries, or changing the batteries (it would be rare, but you could have a defective siren if you know it was set up correctly).  Also check your settings to make sure your siren is set to sound the alarm, and/or if the base station is also set to sound the alarm.

If you have SS2, the siren needs to be synced to the base station separately.  With SS3, it's integrated just like the rest of the sensors.  If you did a user-initiated test, and tested the siren, did you get a call from Monitoring?  Are you getting 'automatic test signals received' in your Timeline of Events? (for SS2, it's weekly on the same day and time - for SS3, it's daily, at the same time every day) - If you did NOT get a call from Monitoring when in test mode, you could have a weak or non-existent cellular connection, in which you may need a new cellular module (i.e., if you have t-mobile, ask SS for the verizon module instead if you think it has better coverage in your area).

If this happens again, or you find any other issue with your system, call SS and request a senior supervisor to assist in troubleshooting.

I only just bought the system, does that mean it will be SS3?

I'm not sure what you mean about integrated, we had to pair all our sensors with the hub one by one. It knew which ones we had because it warns us the ones we didn't turn on yet aren't working, but resetting the thing I'm sure my wife had to go around every single sensor again.

We don't have WiFi connected yet so I wonder if this might be a factor. But how the alarm triggered the dispatch process when the system was in OFF mode is particularly strange. I'd wondered if it's possible to accidentally press the panic button but we can't see we did that.
>Sometimes electronics are weird and just need to be reset from time to time.

Some daft smartplug maybe. a SECURITY system needs to be reliable. Like in your car or an aeroplane, these systems need to be extremely resilient.

@Mister Boy first question: where did you buy your system? Best Buy, Costco (or some other 3rd party) or direct from SS? If Ebay or another site, you may run into issues.  If your keypad shows a WIFI setting, it is definitely SS3; if not, its SS2 and I would, if possible, return it. Are you on a monitoring plan? If so 24.99 or 14.99? SS is a very, very "standalone" product even with some of the integrations they have (Echo for example) and limited so I doubt that would be an issue.

Also, make sure your software updates are current.  In the first early, early systems there were some issues but have long since been addressed.  If you haven't already, I would definitely contact SS support for assistance.

Finally, not connecting to WIFI would not be an issue.

By integrated, I mean, the sensors are paired with the base, just as you did when you set it up.  If you had SS2, the siren was a stand-alone component that had to be synced to the base station separately, by a different method (pressing the keyfob multiple times while at the same time, pressing a black button in the siren's interior).  I believe you have SS3 (as Captain said, if you saw wifi capability in the keypad settings, you have SS3 (can also see the difference here:  https://simplisafe.com/alarm-sensors )

Again, check your timeline of events from your online control panel, it will show which sensor was tripped.

You don't have to have wifi to use SS3, but if you ever want the OTA updates, it requires wifi, which means, you'd have to take your base station and keypad(s) to a friend or somewhere with wifi access to download and apply the updates (so far, SS3 has had 4 or 5 updates since early 2018).