The human-powered land speed record is 82 mph on flat ground; a top cyclist can usually put out about 500-600 watts of mechanical effort for a minute or so.
I recall from somewhere that a pro-level rider may break a kilowatt. Sorry, don't have a citation for that.
The capacity of the battery is only a factor if the total battery power output limits the motor's power output. For example, the 2004-2009 Toyota Prius has a 1.3 kWh NiMH battery rated for a 20C discharge (20 * 1.3 kW = 26 kW). The MG2 motor is rated at 50 kW, but because of the small battery size in combination with the 20C discharge limitation, the motor's actual output is limited to about 25 kW.
Yes. The Prius is limited by the battery internal resistance, which in turn led to a redline in the computer. Toyota doesn't want you frying all those juicy cells. The Ford Escape hybrid, using a similar overall setup, can be spotted by the big cooling exhaust on the driver's-side D-pillar.
I was thinking that one upgrade to the Prius could be a dual battery pack. On a reasonable timescale (~a minute), the computer would switch back and forth between packs. This would give them a chance to cool down, and extend battery-only range, at only a modest weight gain. Of course, this would be less practical on a motorcycle, and worthless on a bicycle, where the extra weight of idle cells is noticeable.
Carlos