REQUEST TAXI TO PARKING: This is done after you have reached the (second) hold short line after exiting the runway.
This is not done while at a full stop on the runway, blocking all inbound and outbound traffic.
Departure Runway: This is determined by ATC (where active) based on a couple variables. They cannot control where you spawn. But the main concern is traffic flow, not giving one, individual pilot the shortest taxi. At an airport like KLAX or WMKK, you may have a long taxi. It’s unfortunate, but there are more things to consider besides your personal expediency. You’re welcome to change your spawn point to somewhere closer.