Electrical – Need advice on UPS circuit with variable voltage input

If I understand you correctly, you want a charger than can switch between battery and input source (i.e. when adapter is plugged/unplugged) to power the load.

There are plenty of charge control ICs and circuits around that can do this. The MCP73831 is a nice cheap little Li-Ion charge IC, and with the addition of a PMOS/Schottky, can switch between adapter and battery. Here is an example circuit:

The output is at the top right where the wire disappears (from the PMOS/Schottky)
Here is an Microchip app note which goes into some details on such a circuit.

The way it works is when the USB power (or adapter) is not present, the VUSB_IN line gets pulled to 0V by R4. This brings the gate of the PMOS (G on the symbol) to ground and turns it on (i.e. opens the Source-Drain, marked S and D) allowing the battery to power the circuit. The Schottky (marked A and K) stops the battery raising the VUSB_IN line and turning the PMOS off again.
when the adapter is plugged in, the gate is pulled to +5V and the PMOS turns off, leaving only the adapter voltage powering the circuit and letting the battery charge.

UPS-powered wall sockets certainly exist in some buildings. At one place I worked at, we had red-colored sockets that were on the building's (giant) UPS (batteries, generators, etc.) and normal/white color for the regular stuff, with the convention that employees would not plug non-critical equipment in the UPS-backed sockets.

You could do something similar to the above at your place if you can access the house's wiring and disconnect certain sockets from the mains and then route them through an UPS.

But what you propose doesn't seem workable firstly because you don't seem to have a plan on how a big consumer like a vacuum cleaner would actually bypass the UPS. If mains power fails while the vacuum cleaner is on, you'd still overload the UPS and cause it to shut down turning everything else off too. (Unless you want to place bets on the mains never failing while you vacuum clean... but I won't delve into that.)

And (if I'm not mistaken on this second part) if you use your UPS to power the mains, when the mains power fails, your UPS will basically try to power the entire neighborhood! So it will probably not work for long... This is one of the reasons why you need an additional device like a transfer switch that Spehro Pefhany mentioned in his comment. In a sense, your UPS contains one of those too because when it starts to power the load from its battery (in case of mains failure) it won't try to power the mains as well. So if you do plug the UPS' output to your house's mains, then you need achieve a similar disconnection effect further upstream in order to prevent your UPS from powering the entire neighborhood. I'm not really familiar with stand-alone transfer switches, so if someone else wants to propose/outline a solution based on that... by all means do so. However the problem discussed in my previous paragraph (potentially overloading the UPS with big internal consumer) would not be automagically solved by an upstream transfer switch, so you'd [still] have to take additional measures to segregate the internal loads.