Your max discharge will be determined by the inverter. So, at 3000W and 15.5V, that's 193.5Amps.
However, I doubt you'd be running at a full 15.5V. Going with 18650 cells, your nominal voltage will be 3.7 * 4s = 14.8V. So, at 3000W that'd be 203A.
Running with that kind of configuration, you would need a lot in parallel to be able to handle the surges from time to time. So, at 1A max per cell, that'd be 200 cells in parallel per pack.
Now, that doesn't mean you have to make a pack that large. You could check your power usage and see that you only ever draw a max of 2400W in a brief moment as the fridge, or washer, or what not kicks on. So, if 2400W is the max, you'd only need to build for 162A draw.
If you haven't bought the equipment yet, I would highly recommend going with 24V. Or, if it's possible to buy a 24V inverter, go that route. That cuts your amp draw in half. So instead of 203A/162A, it'd be 101A/81A respectively of what I quoted earlier.
Charge current is not based on inverter. It's based on what the cells can handle first and foremost. Then it's based on your charge controller. You can't charge higher than what your cells can handle. Different cells manufacturers and chemistries have different requirements. You must check the datasheets to find out what the max current input is for those cells.
With that said, if you build a pack that can handle 100A discharge, and you have 100 cells in that pack, each cell with discharge/charge at about 1A each if you charge with 100A.
What ever your input charge/discharge current is, it will be divided by the number of cells you have in the parallel cells, not series cells.