As soon as the plane has landed, the "flight information department" receives flight parameters, including "overload" (as an example), what other parameters may indicate the correct operation of the pilot?
If you are referring to commercial airline operators, then you most likely meant a Flight Operations or a Flight Safety department receiving this information. Most airplanes do not automatically send all of the information to the headquarters but rather store it in an onboard storage device, then this information is downloaded from the aircraft during scheduled maintenance (every 3-4 days). However, newer aircraft can and will transmit any abnormalities (e.g., high oil pressure, engine fire) directly to the company in real time.
As far as what parameters are used, there are too many to list. Most important values that the airlines look for include the rate of descent, approach speed (ensuring that it is appropriate for the phase of flight), flight path and aircraft altitude over certain waypoints, and the use of flaps/landing gear. All of this is covered under the Flight Operational Quality Assurance (FOQA) program. Fun fact: Airline management have only access to de-identified information, meaning that the data does not show the date of flight, pilots' names, and any other information which could identify who flew a particular aircraft on a particular flight. The purpose of FOQA is to analyze the operation as a whole rather than look into each individual pilot. The latter is completed during pilot proficiency checks which are conducted regularly.