Since this is a particularly confusing part of measuring batteries, I'm going to discuss it more in detail. Power capacity is how much energy is stored in the battery. This power is often expressed in Watt-hours (the symbol Wh).
The number of batteries required to power a house depends on the size of the battery you choose and the appliances that need to be powered. The larger the capacity of the battery, the fewer batteries you'll need. You'll also need to take into account your home's energy consumption and what you plan to use the battery for.
Occasionally lithium battery cells are marketed with just a C rating and not a maximum current rating. This can make it easier to compare the power level of battery cells of different capacities. As long as you know the capacity of the cell, you can use the C rate to quickly calculate the maximum current rating of the cell.
So, yes. Batteries have a max current drain (given by design and physical/chemical limitations) and yes the storage rating (being Ah, Wh or Joules) changes depending on battery design and load applied, and yes Wh is a better way to compare batteries because it takes voltage in account.
Power capacity is how much energy is stored in the battery. This power is often expressed in Watt-hours (the symbol Wh). A Watt-hour is the voltage (V) that the battery provides multiplied by how much current (Amps) the battery can provide for some amount of time (generally in hours). Voltage * Amps * hours = Wh.
However, the amount of current we can really draw (the power capability) from a battery is often limited. For example, a coin cell that is rated for 1 Ah can't actually provide 1 Amp of current for an hour, in fact it cant even provide 0.1 Amp without overextending itself.