Welcome to Islam.SE. Thank you for your question ... Since you have asked 3 questions , my answer might seem a little long. But I beg you for your attention.
As I go along I will also give you the chapter and verse number from the Quran so you can verify it easily.
Firstly, since you have not read the Quran, I will paste some verses from the Quran and I leave it upto you to decide if they are violent or not.
Who spend [in the cause of Allah] during ease and hardship and who restrain anger and who pardon the people - and Allah loves the doers of good; (Quran 3:134)
"And the servants of the Most Merciful are those who walk upon the earth easily, and when the ignorant address them [harshly], they say [words of] peace," (Quran 25:63)
"Because of that, We decreed upon the Children of Israel that whoever kills a soul unless for a soul or for corruption [done] in the land - it is as if he had slain mankind entirely. And whoever saves one - it is as if he had saved mankind entirely." (Quran 5:32)
"And whoever is patient and forgives - indeed, that is of the matters [requiring] determination." (Quran 42:43)
"There shall be no compulsion in [acceptance of] the religion. The right course has become clear from the wrong..." (Quran 2:256)
Q1: Is Quran saying that other non-islamic religions should be destroyed/punished/, even killed may be?
Surely those who believe, and those who are Jews, and the Christians, and the Sabians, whoever believes in Allah and the Final Day and does good, they shall have their reward from their Lord, and there is no fear for them, nor shall they grieve. (Quran 2:62)
Normally it is very unlikely for a religion to testify for another religion BUT in the above verse the Quran says that Jews, Christians & Sabians will be rewarded for the good they do and the faith they have.
I have not come across a verse that says kill people of other religions. If you have a quote, please feel free to open another question for clarification.
Q2: When is it ok (if ever) for a muslim to fight back or act violent open another human being?
Fight in the way of Allah those who fight you but do not transgress. Indeed. Allah does not like transgressors. And kill them wherever you overtake them and expel them from wherever they have expelled you, and corruption is worse than killing. And do not fight them at al-Masjid al- Haram until they fight you there. But if they fight you, then kill them. Such is the recompense of the disbelievers. (Quran 2:191)
The above verse is in the context of war, but even in that context you will notice this verse talks about fighting as a defense BUT within boundaries.
Q3: For example Jesus is quoted (see below) saying that once you have been hit on the left cheek, ...... Does a rule like this exist in Quran?
I have already quoted many verses from the Quran that would suffice but I wish to explain a little more about such verses.
The teaching of the bible "turn the other cheek" is a very good teaching, but in my opinion it wouldn't always work in todays times.
For example : If a some one breaks into your home, and after he is caught by the police, would you invite him to break into your home again? NO.
As I have already shown the Quran requires us to be patient BUT it also requires us to be JUST. So when patience is required, we have to be patient, but at the time of justice we have to make sure justice also gets served.
"O you who have believed, be persistently standing firm in justice, witnesses for Allah, even if it be against yourselves or parents and relatives. Whether one is rich or poor, Allah is more worthy of both. So follow not [personal] inclination, lest you not be just." (Quran 4:135)
In this verse the Quran says you should testify against yourself if you have done something wrong for justice being served.
Beware of Sound bites
There are many people (muslims and non-muslims) that quote the Quran out of context for personal gain. So please beware of such misquotations that cause misconceptions. They become very popular without even verifying with the Quran.
I would request you that if you have any questions about Islam, please feel free to post it on Islam.SE where you will get references to the Quran which are even easy to verify.