That is a very complex question. What do you mean with ATC facility, exactly? ACC, sector control, approach or something else? Their operation will depend heavily on the type and if it is US, EU or somewhere else.
I can tell you, that there will not be any conflicts between a pilot and an air traffic controller, everybody is very professional, if a pilot really does something wrong or does not comply, a report will be issued, after the fact.
An ACC may receive an emergency call and re-route the plane accordingly. But typically they won't handle emergencies, as the vast majority happens during take-off.
En-route conflicts will not occur in a managed sector, if there should be one unexpectedly, the pilots are responsible, as a last resort you have the TCAS. At least in europe, that is the order of precedence.
Sectors are split according to a fixed plan. Controller roasters are similar to pilot roasters. You can't change them on a whim. Controller workload is watched closely. Also the errors they may make.
Noise reduction is a concern of SIDs/STARs (the routes close to an airport) of the given TMA and they are not in the purview of either an ATC controller or supervisor. They are issued after a lengthy process by the ANSP.
This is a general answer for the situation in EU/US. Things may be different elsewhere.
The ATC facilities:
In a tower, the supervisor will usually
- ensure safe conduct of the watch
- check flight plans, slots, etc
- liase with centers, cfmu, airlines, etc
- handle, support, supervise emergencies
- day-to-day management
- controller management (breaks, time, etc)
Depending on the airport controllers will occupy a wide range of positions.
Approach is typically a different facility, that is colocated and has their own supervisor. The tower supervisor will coordinate with the approach supervisor.
The ACC has similar duties to a tower. The difference is, that the positions are for one sector, there are much more positions. There may be two controllers per sector. You will then have 1-2 "supernumerary controllers" for X controllers.
Apart from that you will have a number of assistants and other personnel for a variety of duties.
In an area control center, the supervisor will ensure, that all this works well together, so much more responsibility.
In this situation, we are in the tower on ground control. There is not much a pilot can do other than follow the instructions. Sometimes they have to wait, sometimes they are forgotten or they loose their slot. Sometimes the lineup is slow, there are a number of things that could happen.
It should be noted, that most airports have a departure management software to de-conflict aircraft management. Most pilots know that.
Here a supervisor might go and ask, if a plane can be squeezed in, if it lost its slot.
speed / altitude
In controlled airspace, a pilot will follow the instructions to the letter. There might be communications issues, "climb to" / "climb trough"
If they don't comply here, that would be a major issue and would trigger all sorts of alarms. It could ultimately end with two fighter jets next to the aircraft...
The controller would handle the aircraft and emergency services, the supervisor will call approach, they will divert all aircraft, most likely back in stacks, then he will close the airport for traffic. Every airport has an ATIS, the message there would be updated. Then the ACC is informed and also cfmu/network manager.