The close door button in most lifts doesn't do anything at all, in fact it's often just a piece of plastic stuck on the wall with glue. Similarly the button at pedestrian crossings doesn't always do anything at all except turn on alight that says wait. The ones around West way in Barnsley are a good example of this. Press them all you want but it won't change how long you wait.
The pedestrian crossing will stay at green until someone presses the button to signify they want to use it though.
Depends on the crossing. If it's just on a plain old road then yes, but if it's a junction it's a placebo
That depends what kind of crossing it is. Everything around Barnsley town centre runs on a timing system to keep the traffic moving so pressing the button does nothing or it would disrupt the entire flow of traffic throughout the town. Its the reason the lights continued to go through their sequence when the road was closed for a couple of weeks following the Chicago rock fire rather than being set to red or turned off
I probably didn't explain it very well first time round. The random ones on a straight stretch of road do something and some of the ones I described do too in the middle of the night but as a general rule the linked up type are just there to make you feel in control.