Some interesting ideas from Solarman but it may not be quite such a good solution for the following reasons.
1) Low cost inverters when used 24/7 are not very reliable as they are not designed to run continuously day after day and will probably fail due to their capacitors drying out in 6-12months. Inverters can be purchased that will last but they are not cheap.
2) Low cost inverters have very little internal energy storage and therefore draw large 100Hz haversine ripple currents, now one of the things that shorten lead acid battery life is ripple current and as Corporal Jones remarked "they don't like it up um".
3) To keep the charging ripple current low and well regulated a simple low cost battery charger will not do the job it will require a well regulated output and as this will also see a proportion of the ripple current into the inverter some decent output capacitors are required so you need a decent regulated power supply again not cheap.
4) In the event of the inverter failing there is no backup, in a dual conversion UPS if the inverter fails for any reason the output is connected within a few ms to the incoming mains so the load is supported and an alarm is raised allowing the failed UPS to be sorted out. The lack of a static switch which can select inverter output or live mains is critical for system reliability.
There are three main types of UPS:
1) Simple: Mains powers the load via static switch and in the event of the mains failing the load is powered from the batteries via an inverter, in the cheap models the output from the inverter is a square wave. A small battery charger keeps the batteries topped up when mains is available.
2) Line interactive: Similar to the above but the UPS has a tapped transformer and a means (relays usually) of selecting the appropriate tap so the output of the transformer is kept within limits.
3) Dual conversion: Mains is converted to high voltage DC and an inverter converts this back to AC, In the small models the DC rail is usually around 350-420V and sufficient energy is stored in capacitors to allow time for the low voltage to high voltage DC-DC converter to run up without a break in mains output.
For the static switch to do it job the inverter output and the mains need to be synchronised.
Next question, Ebay item 400218096566 should do your job OK. As stated in the advert it is Sine Wave output, from memory these are Line Interactive units (see above) and although not quite as good as a Dual Conversion UPS it will power the devices you listed as all the devices should have 20ms of holdup within their power supplies.
Dual conversion and Sine Wave UPS will generally state such in their specifications.
If you want to ensure your computer shuts down properly even when you are not present then using the USB or serial output from the UPS along with the software provided should ensure you keep your data intact.