Very interesting, thanks for posting.Get your units right first and that might help.
Energy is measured in Joules (becomes relevant later). One Watt (capital because it was named after someone) is defined as one Joule per second (again capital for the same reason). Kilowatt (kW) means 1000 Watts or 1000 Joules per second.
This only tells you the rate at which you are using energy not how much you have used. To know that you need a time frame. Hence kWh (kilo Watts per hour).
An example, it takes 330kJ of energy to boil one litre of water. Therefore if you had a 330kW kettle you could boil 1 litre in a second. Realistically you don't so a kettle is say 2kW. This would then take 330/2 or 165 seconds to boil the water, 2 minutes 45.
Now to work out how much electricity you've used in conventional units you need to convert them to the same units. We've used 2kW for 2m45 so that's 2x(2.75/60) = ~0.1kWh (note we have to divide runtime in minutes by 60 to get it to hours).
Battery capacity tends to be quoted in either kWh or Ah. So a 30kWh battery can supply 30kW for one hour or equivalent. So 15kW for 2 hours and so on. Note depending on the design of the battery it might not be able to deliver 30kW, it might be limited to a maximum output of say 2kW in which case you can max it out for 15h.
Using your example of a 30kWh battery over 24h then it would be capable of delivering a continuous 1.25kW for 24h. If you only had 100W load for part of the day then you can draw more at other periods.
If you think of it in terms of a graph with kW on the y axis and time on the x axis then the area under the graph can't exceed 30kWh.
If the battery capacity in Ah then you need to know the nominal voltage because Voltage x Current = Power (V x A = W) to get the capacity.
Bear in mind that an inverter won't be 100% efficient so you lose some power there.
When you get into the nitty gritty you need to start looking at graphs rather than "figures" as the capacity isn't linear with voltage etc., so to get a more accurate picture you have to start working it out for different profiles etc (adding areas under graphs or integrating).