The satellites' location is very accurately determined using ranging techniques, where a pulse is sent from a ground station to the satellite and the satellite responds very quickly, with a known time between receive/transmit. This allows one to determine exactly how far away the satellite is. Combining this information with orbital tracks allows one to determine exactly where the satellite is.
The amount of time required to do a time update from the time the signal is sent to the time it is received and in use is also well known. What is actually sent to the satellite is the time deltas, to tell the clock to jump forward or backward a certain amount of time. For GPS, this update can occur every 8 hours, but in practice is done every 24 hours.
The satellites each have onboard atomic clocks, for very accurately determining the time. Relativity is managed in essence by correcting for the clock based on the expected time. They slowed the clocks down to correct for this effect.
Signals are sent up to the satellite via the GPS ground network.
The time standard used for GPS clocks is a whole number of seconds from UTC. GPS time does not account for leap seconds, thus the whole number of seconds time difference.