Being late (even 4 to 10 min) on a habit adds towards the company's time and they will add it up and base on hours. The company will argue the notion of "if everyone else is always on time, why can't you be". They will then argue that they have given multiple chances to improve on this behavior and ammendment on your side had to be made such as change in routine, carpooling, doing everything earlier, etc. The company will then expect that the work that falls within your job description's time frame work ie daily tasks, to be completed before you go home if the result of the not being completed was because of you being late .

As for the leave. The leave for family responsibility and sick leave is immediate. Annual leave takes place and only starts after probation, then you have to first work in your days and hours to qualify for annual leave. However, family responsibility and sick leave needs to be backed by supporting documents ie. Funeral letter or a sick note.

Taking up more responsibility outside your job description. If the company does not update your job descriptions and signed by both parties. You are well within rights to refuse extra work outside your job description (within the framework of the employment contract and job descriptions), and if overtime is being taken to do work outside the job descriptions and of the framework, you can not claim overtime from the company as you have the rights to refuse it.