Cutoff: Not lower than 11.8V
Charging time: Charging time depends on chemistry, 7-9 hours since 0% to 100% SOC, and keep in mind it is not linear. Last 10% need 2.5hour, last 20% - 3.5hour and so on. Approx table:
https://www.researchgate.net/figure/Charging-test-result-for-12V-7AH-battery-using-the-developed-battery-charger_tbl1_325627377
This is very good charger, and does it 7 hours at relatively low ambient temperature. Most of UPS dont do good charger job and built just for short interruptions.
Why diesel: Diesel lifespan is much longer, less maintenance, less toxic fumes(not sure about that in Lebanon, due lack of regulations on fuel and exhaust), less flammable, less overheating. But they dont scale well to small sizes.
Load: They oversimplify. It is better to keep load within 60-90%. Underload cause soot buildup, overload cause overheat and higher wearout.
https://en.wikipedia.org/wiki/Wet_stacking
I had very small 1kva petrol generator with inverter, just to keep pc, internet stuff on and charge ups. It has low load mode and run very silent in it
Automation: Most of them have options for remote start and shutdown, IMO it is trivial, if it have it. Sure if you are good with electricity.
3KVA: For me - on the edge, depends if it will handle rush current. Large A/C, power hungry home appliances might require more. You can just install some logging current meter and check log of 1 day
Keep in mind, if you need good place to install it. When you will run it, likely on blackout, people many people will be without electricity too, with open windows, as they have no A/C. They might force you to shut it down, if they smell exhaust.