What is the ruling on the work of an engineer who pays a portion of his salary to the employer as office rent, and does not receive adequate compensation for overtime hours? And is it permissible for him not to give the employer half of the fees he earns from private work, even though he had agreed to that?

If you are a private employee, your working hours are the property of your employer, and it is not permissible for you to work for anyone else without his consent. He also has the right to share in your income. As for your working for someone else outside of working hours, if the office rental includes its benefits after working hours, then the employer has no right to your private work, and you may use vague language regarding the amount of income. However, if the rental only includes working hours, then you may not work [for anyone else] except with his permission, and he has the right to share in your income. Regarding overtime pay, if he does not give you your due or if you do not reach an agreement, you may take what is equivalent to the customary wage from his money without his knowledge. The disagreement on this issue has been elaborated upon.