The Flic App(s) and SDKs have no limitations in amounts of simultaneous connections. The bluetooth implementation of the device you're connecting to sets the limits, and this varies a lot. It depends not only on OS and make, but on Bluetooth chip and stack as well.
Each Flic requires an active connection. Beacons do not require a connection, as they are simply advertising.
We never recommend using Flic buttons as alarm buttons where you would be dependant on it not to fail, no matter what. If you will do this in any case, I would definitely recommend using a RPi.