Bypass capacitor in 555ic
pin 5 of the 555 is &quot;control voltage&quot; a tap on the divider chain that sets the levels for setting and resetting the internal flip-flop.
without a capacitor there the thresholds will tract the supply voltage exactly ad a glitch on the supply (a brief spke or dip) would cause the threshold to move leading to early triggering of the flip-flop - so without the capacitor on pin 5 supply noise can cause your 555 to run fast.
Putting the capacitor there stiffens the voltage there with respect to ground, and if the timing capacitor is also ground referenced gives a large amount of immunity to supply glitches,
so to answer your questions:

Will voltage/current changes in line affect on pulse width?

yes absolutely, depending on the timing of the voltage changes they
may cause the pulses to begin and/or end early

Will capacitor at pin 1 discharge late?

Pin 1 is ground, it's unclear which capacitor you're asking about, because they all connect to ground.
pins 2 and 6 are the timing pins, the charge/discharge of the capacitor there can be triggered early by glitcing the power supply.
it can also be delayed if the glitch goes the opposite irection but that effect would be a smaller effect.