Load regulation and battery charging
The load regulation specification tells you something about how much Vbat changes over different output currents when the battery is charged and the unit is powered from an external voltage. It is maximum 0.18 % (0.08 % typical) of for example 4.2 V = 7.6 mV (max)
 ===> I would not worry about it :-)

No the chip does not expect the battery voltage to increase monotonically while charging. It does not have to. Typical charging mode for Li based cells is current limited when Vbat is below a certain voltage, Voltage limiting above this voltage and charge to a specified voltage. This is exactly what the MCP73871 does.

2000mA is on (maybe over) the edge of what this chip is designed for. The MCP73871 will not shut off I think as it will have to keep the internal switch closed. But it will not drain the battery much, so you can assume it uses no power. This is what it is designed to do :-)