Is it permissible to combine prayers (Dhuhr and Asr) either by advancing or delaying them, or to pray sitting in a Western country for study purposes, when it is difficult to perform the prayer at its proper time and place, based on the fact that the Prophet (peace be upon him) "prayed Dhuhr and Asr together, and Maghrib and Isha together, not out of fear, nor travel, nor rain"?

It is obligatory to observe prayer at its appointed time, with all its pillars and conditions, without shame or shyness, because displaying religious rituals and taking pride in one's religion is religiously required.

Delaying prayer out of shyness from people seeing you is not a legitimate excuse. As for combining prayers for the sake of a school exam, one must go out to perform the prayer at its appointed time if that is possible without missing the exam. If it is impossible to go out and it would result in harm, then there is no harm in combining the prayer with the one that follows it, if they are among those that can be combined.