MCP 23018/23S18 for LED and Switch control
Yes. LEDs in series share the current, divide the voltage. Choose the resistor based on the combined Forward Voltage of the series LEDs.
Vout is not the right term. It's Vss, or Gnd. But yes, 400 mA is the max current that can go through the Ground pin, through the Open-Collector Outputs. 6 LED strings, at 20 mA each is only 120 mA, only a quarter of the max. A resistor is required per string. Trying to just match the voltage is not good.
The inputs are high impedance and not open-drain. They should only draw microamps, or fraction of a milliamp.
All 6 leds and 10 inputs at the same time, sure, as long as you keep within the specs.
The Clamp currents are the max current that can go through the clamp diodes, at a voltage higher than VDD or lower than VSS. Not relevant to your situation.
Current Sunk means current flowing into the device, typically through the open-drain output, through VSS/Gnd. Current Sourced is from VCC, through the voltage high output.
You can do up to 400mA, if your Power Dissipation is under 700 mW, so 500mA isn't significantly better. At 25 mA max per channel, when you want 20mA, you can already do 16 channels of 2 leds, per MCP. A shift register and a SPI device like the MCP23S18 are functionally quite similar as is. As for 500 mA through VIN, the difference would be wiring the in a common anode (To V+) or common cathode (To Gnd) manner. If you are using regular two pin LEDs, then that's not really a concern.


Typical Setup for LEDs: