Is the Pope morally obliged to submit to every secular authority? Among the Protestant faith a believer is to submit to the government:   


  Submit yourselves for the Lord’s sake to every human authority: whether to the emperor, as the supreme authority, or to governors, who are sent by him to punish those who do wrong and to commend those who do right. (1 Peter 2:13-14, NIV)


I am unsure about medieval historical details but I had the impression that there were struggles between the Catholic church and  the state. I have the impression that the Pope tried to argue supremacy over the state in opposition to the words of Peter.

My main question is not, 'Did the Pope really try to do this?' My question is does the modern day Pope teach his duty to submit to secular authority? 

According to catholic teaching is the Pope morally obliged to submit to every secular authority?
It's a mistake to interpret the 1 Peter passage as meaning "every conceivable human authority". It means only those that have legitimate authority over you. People in Roman Israel were not expected to obey the laws of Roman Britain, unless visiting there. Likewise a US citizen is not expected to obey the laws of France while in the US.

Christians have also recognized that there are situations which override this passage. If the state forbids you to worship or follow God, that is not something you should obey. If the state commands you to kill and persecute those of a particular race, that is not something you should obey. Jesus himself demonstrated this by refusing the commands of the religious authorities to stop teaching.

On the specific case you ask about, the Pope is head of state of the Vatican, and as such is his own 'human authority'. That doesn't mean he is immune to the rules of that state, as is the case with most heads of state. If he were to violate the rules of the Catholic Church he would, as I understand it, be subject to discipline.

Regarding other countries' authorities: countries rules mostly don't extend to those who who are not citizens and not present in the country. Visiting heads of state are usually subject to diplomatic immunity, and so other countries' rules again don't apply.

In the unlikely event that the Pope found time to study for a fine arts degree, and that he didn't do so at a Vatican institution, I don't see any reason to think he wouldn't abide by the rules of the institution. 

Finally: the Pope has only been in his office a week. If you want to find out what his teaching is on this subject, I recommend waiting.