Looking for ideas for a 5V power supply with 12V battery backup
You have two separate problems, charging a 12 V battery from 110-220 VAC, and creating 5 V from the 12 V.

The first is easily solved with a off the shelf power supply with universal input and that puts out the right float charge voltage for your battery.  Most power supplies nowadays have universal input, so that part is easy.  Most lead-acid batteries are fine with being indefinitely held at the float charge voltage, which is a bit over 13 V for a "12 V" battery.  You can probably tweak up a 12 V power supply to make 13 V or so, or you may be able to get a power supply intended to charge and maintain a 12 V lead-acid battery.  Those are usually sold as battery chargers instead of power supplies.

As for converting 12 V to 5 V, that's a well solved problem so no need to go into detail here.  For such a voltage drop, it should be a switcher.  You should be able to find off the shelf DC-DC converter modules that can do this, or make your own with any number of buck converter chips.  There is also a drop in buck switcher replacement for the old 7805 linear regulator.  Again, there are many well trodden paths here.