From what I understand, an aircraft's registration number is its ICAO code in the pattern found here: Wikipedia: Aircraft registration, but I have seen references to aircraft using a 3-letter code (I'm assuming it's an airline code) followed by a 4-digit number. (Example: PAK0032) What kind of a number could that be?
That format is the flight id. The first three characters are the ICAO operator code which identifies the airline. The rest of the characters are unique to that flight. The first character after the ICAO operator code is always a number, after that both numbers and letters are allowed. The maximum length of the flight id is 7 characters.
The flight id is transmitted to Mode S radars by the transponder and also broadcast on ADS-B. It is used to correlate a flight to a flightplan.
The flight id is also the basis for the callsign used in radio communications. The ICAO three letter code represents the airline's callsign, the rest of the code is pronounced in the ICAO alphabet.
The registration for an aircraft follows the nation's letter assigned by ICAO, correct.
But this is not the only instance where ICAO is used. When the registration is not used as the radio callsign for the aircraft, They will use the flight number to address the aircraft on the radio.
The flight callsigns do not always follow the flight number you see on your ticket, especially since the radio callsign uses the ICAO code for the airline instead the IATA code on your ticket. Some airlines even replace parts of the flight number, either to avoid similar callsign at the samw time or to obfusticate which flight it is.
LH4892 could be known to ATC as DLH48RA
Some transponders can be programmed to broadcast the flight number if they operate with ADS-B. In these cases they carry a specific hexadecimal value, which can look similar to the code you posted, but can have the hexadecimal values in any place and not only the letters in front.