Is the use of government work computers and electricity for Islamic proselytization (da‘wah), publishing religious websites, and listening to the Quran outside official working hours or when there are no tasks, considered a violation of a right that does not belong to the employee, and what is the expiation for that if it is so?

It is permissible for you to use the internet and equipment of a government institution for dawah (inviting to Islam), if the institution does not incur additional financial cost for this use, and has explicitly granted you permission, provided that it does not harm the interests of the patrons. If the institution does not grant permission, or if it results in an additional cost, then it is not permissible for you to do so. Inviting to Allah is a noble goal, and the means to achieve it must also be noble. If the use is without permission, then you must seek forgiveness from the officials or pay the resulting costs.