Is it permissible, Islamically, for me to do the work of lawyers in writing petitions and legal studies, and to receive a fee determined by the lawyer for that?

Your cooperation with lawyers and writing petitions, alongside your work at the company, is a cause for accusation and suspicion of favoritism towards lawyers or negligence in overseeing them. An employee is a private hire whose employer owns his services during official working hours, and he is not permitted to undertake private work for anyone other than his employer during this time without permission. Therefore, you are not allowed to do so except with the permission of your employer.