Regarding Question 1:
NMR measures the energy difference between the field-aligned energy of a nuclei and the field opposed energy of the nuclei.
The biggest field in the situation is the external magnet.
The next biggest field is the magnetic moments of the electrons. Electron density leads to what we refer to as chemical shift.
Lastly, there are the magnetic moments of other nuclei. But, the other nuclei are also undergoing their own spin flipping and such like. So if a hydrogen has its molecular neighbor with a spin aligned with the external magnetic field while relaxing, the total field it experiences is stronger. Otherwise it is weaker. The difference between stronger and weaker is essentially the distance between the two peaks on the NMR spectrum. It is important to remember the external magnetic field is the strongest here, so when the nuclear spins oppose and weaken (or align and strengthen) the magnetic field, it is only a tiny amount.
Regarding question 3:
Carbon and Oxygen have complete Nuclear quantum shells, and as a result have no net magnetic moment. The concept of 'spin up' or 'spin down' means nothing to these nuclei.
Since splitting is the result of a magnetic interaction between spins, there will be no splitting.
Regarding chlorine, The coupling constants are small, and generally not observed due to limitations in the machines making the measurements.