I ordered "Replacement battery suitable for APC Back UPS 550, 650, 700 replaces RBC110" and it has been working perfectly again since then. But it wasn't available from Digitec. But if you google it you should find it.
Hello and yes, it does play for the Syno absolutely. For the shutdown signal, you can try to connect a USB cable from the Nas to the APC and configure in "Systemsettings/Hardware&Energie/USV - IF the Nas recognizes the USV, it is never guaranteed because APC and its protocol is proprietary and can change...
Sincerely and happy holidays
To 1:
The sine wave is "stepped", i.e. simulated. The question is not what comes out, but whether the connected devices need a real "pure" sine wave at all.
To 2:
From my point of view, you can connect everything to one output. however, this makes no sense if you have 4 outputs available. the total output power according to the product specification must not be exceeded!