Is the questioner considered a disobedient daughter to her mother or negligent in her duty towards her, given the circumstances her mother went through of illness and death, and that the questioner was studying far away from her, despite her attempts to reconcile between her studies and caring for her mother as much as possible? What should she do to get rid of her guilt?

What you did towards your mother, in terms of offering advice, is not considered disobedience (ʿUqūq), even if there was some overstepping, as long as you are good and dutiful towards her. Allah Almighty says: (Your Lord is most knowing of what is within yourselves. If you should be righteous, then indeed He is ever Forgiving to the often returning [to Him].) (17:25). You can expiate for any shortcoming by supplicating for your mother, giving charity on her behalf, honoring her friends, and maintaining ties with her relatives; for this is a form of dutifulness that does not cease after her death.