Does the eeg tracing change if the patient is thinking about different things ?

Not really: Eegs are nonspecific and mostly helpful to detect seizure like activity. Awake and varying levels of sleep are also reliably detected. Thinking different types of thoughts may not be reliably detected.

No: An eeg records the surface electrical activity of the brain. The eeg changes with structural or metabolic abnormalities and sleep cycles (including early sleep stages which you are not completely asleep). Different thoughts do not lead to noticeable changes on eeg recordings.