The comparator doesnt care about amplitudes at all. It only cares about the instanteneous voltage at the two inputs. It doesnt know that the two inputs are some waveforms, it just compares the input voltage all the time.
Without the pullup, the output can only sink current because it has only a transistor between ground and the output pin, so you dont get any voltage output unless you apply some voltage to the output. Read something about open collectors before you ask again.
Opamps are meant for amplification with high gain. When you let the opamp output to swing hard to either end of the possible output voltage, the transistors inside saturate and transistors take long time to get back out of saturation, so in effect the comparator made of an opamp is very slow.