What is the ruling on counting the doubled work that was performed to cover for a frequently absent colleague, in exchange for deducting personal days of absence, which are far fewer? And how should the form for monitoring overtime hours be dealt with in a way that pleases Allah?

If you reported your colleague's absence to the administration, you have fulfilled your duty. You are entitled to claim compensation for the extra work you performed. If the administration does not respond, it is permissible for you to falsely record extra hours to claim your right, because lying is permissible if it is the only means to obtain a right, as mentioned by Ibn al-Qayyim and Ibn al-Jawzi, provided that it does not result in harm.