LGW 15
1. When the aircraft is ready the pilot will talk to ATC on the appropriate radio channel to ask for permission to start engines and push back from the stand. ATC will give permission when it is OK. The pilot is also in contact with the ground staff and/or tug driver via a kind of intercom system and can relay intentions and any ATC instruction received. When that is all done the ground staff will disconnect the intercom cable [which you may see...it is normally plugged in somewhere near the front of the aeroplane ] and walk to a position so that the pilot can observe a hand signal to confirm that all the ground equipment and people are out of the way. The pilot can then request permission to taxi from ATC.
2. The airline will know in advance which parking gate has been allocated by the airport authority. The ground staff will be ready to meet the aircraft....either ATC or the airline company will have radioed the parking position to the pilot. Last time I worked at an airport I recall that an airbridge operator had to be correctly qualified to "drive" the bridge...it is a responsible job because it could so easily damage the skin of an aircraft.
Hold out a finger at full arm-stretch, and a finger of your other hand about half the distance away.....move your head from left to right and back and notice how the fingers line up one behind the other.....well the pilot can use a system of pre-positioned lights or markers in much the same way to assist parking in the right spot for the airbridge....or sometimes a marshaller will do the same job.
I apologise if I've oversimplified.....I don't know what you know and therefore I answer with due respect.