I am wondering if there are certain rules about when a pilot and when a co pilot starts or land a plane. As far as I understand a co pilot's job is to support the pilot. Do they talk to each other during the flight and decide that spontaneously or is the decision already done long time before a flight and how?
Before the flight starts, one of the two pilots is always the "PF" (Pilot Flying) whose job is to fly the aircraft, and the second pilot is "PA" (Pilot Assisting) whose job is to deal with the communications, aircraft systems and perform tasks as requested by the PF to assist them in flying the aircraft.
It is usually the captain that does the preflight briefing in the cockpit, announces any anomalies for the aircraft (for example, the APU might be unavailable due to maintenance, etc.) and all this is communicated to the rest of the flight crew.
Then, it is decided who will fly this leg of the journey, and that pilot assumes the control, and the other starts communicating with the tower/ground/dispatch and adjusting the various systems, etc.
The ultimate responsibility rests with the captain and if there is any abnormal situation (such as a flight systems malfunction) or other condition, the captain can always assume the duties of "PF".
The exact manners on how this is done differs, but for the purposes of the flight log it is always annunciated. So the PF might say "you have control" and the PA might say "I have control" to indicate that the flying roles have switched.
Crew management is a complicated topic and is a complete discipline and many airlines spend a great amount of time training and teaching crew management to their pilots.
You've asked a many-faceted question, the answers to which depend a little on FAA regulations in the U.S. but mostly on the policies and culture of individual airlines. Also, it's the captain's airplane, which means he/she may have, should we say, his/her own way of doing things. How much that might vary from the airline's "standard" can vary greatly, especially when you compare what would be typical in, say, the U.S. versus other countries, and especially third world countries. You could write a small book on this topic. A few items I can think of offhand follow.
All four of the airlines I flew for (two commuters and two 747 carriers) required the captain to do the engine starts. These days there is FADEC, so it's an automated procedure. Pre-FADEC, somebody had to move the lever that introduced fuel to an engine and, in the case of a hot-start, had to cut off the fuel and handle the heat properly. Failure to do so meant you might trash a multi-million dollar engine.
At all four of my airlines, the usual procedure was for the captain to fly the first leg and then alternate with the f.o. after that. At the two 747 carriers, however, new captains were required to do all the landings for their first 100 hours.
The second commuter I flew for had a rule that when the nose-wheel steering of the aircraft (SA-227 Metroliner) was inoperative, the captain had to taxi the aircraft.
The second 747 carrier I worked for had a rule requiring the captain to do all the taxiing. I ignored that rule as I felt it was unnecessary and ill-advised.
There were rules that required the captain to do approaches when the weather was below certain minimums.
The standard way of doing things was always well covered in initial and recurrent ground schools, simulator training, and CRM (Cockpit Resource Management) classes and sim exercises. IOE (Initial Operating Experience) for new cockpit crew as well as later periodic line checks by check airmen provided continuing standardization.
During a flight, the pilots continually talk to each other as required, and on long flights typically a fair amount of visiting, although below 10,000 feet only essential conversation is the rule. There is an old joke that goes something like, "When pilots are in the cockpit, all they talk about is women. When they're with women, all they talk about is flying." Obviously it's a sexist joke and is, I hope, disappearing with increasing numbers of women in the cockpits. It was never really true anyway.
Concerning verbal communication, both 747 carriers, as a part of Cockpit Resource Management had a procedure for handling subtle incapacitation of a pilot. If at any time you did not get a verbal response from your fellow pilot when a response would normally be expected, you were to turn to him and say distinctly, "Are you okay?" If they didn't answer, you were to take control of the aircraft.