The abs pump will start as soon as the ignition is turned on if there is not enough pressure stored already(in the sphere you can see on the side), until it reaches the required pressure as determined by the pressure switch on the abs pump unit, this from overnight can take some 30 secs on some.
When up to pressure it will extinguish the abs light and the handbrake/low fluid level light.
Anytime the bas light stays on is due to a partial or complete malfunction of the abs, although you will still have the brakes, the effort required may be a lot more.
I've never heard of a pedal vibrating when pressed when stationary(?) but unclear whether you mean driving or stationary ?.
The pump is there purely to provide pressure to assist(like a servo), and plays no actual part in the abs function other than this, but if you pump the pedal enough it will have to build more pressure as some has been used, but certainly should not be continuous.
How hard, "stamp" on the pedal should not make any difference to the pump coming on, as there should be plenty enough there.
I cured my abs light coming on a few weeks back, by completely de-pressurising with ignition off, then turn on and allow it to pressurise a few times, seems my pressure switch may be a bit lazy, so might be worth a try.
The pedal should be nice and firm, but not solid, when you say the brakes are poor do you mean as in bite, or a massive effort needed ?
tabetha