Does Islam have to be political?
Islam has both political and spiritual aspects. Whether Islam gets political or not is decided by Muslims themselves. Islam in some parts of the world is less political (e.g. Europe, Russia) and in some other, highly political (e.g. Middle East, Pakistan).
It seems to me that there is no country with a muslim majority where
Islam is not used for political reasons. All of them have at least a
few laws forbidding sex before / out of marriage, alcohol, leaving
Islam, or a muslim woman marrying a non muslim. Why is the state
watching over this behaviour, and why is it not left to the people
themselves to behave?
Most of these countries are democracies where people vote to decide who they will vote for. So, your assumption that people have no say in those decisions in not true. Various surveys (e.g. Pew) have shown that a significant portion of Muslims support those laws and restrictions.
It seems to me that forbidding something by fines or jail time leads
to a lot of hypocrisy. Not only from the people, but from the rulers
too, and even more so since they can use islam for their own material
gain. Is there any part of the quran (or hadiths, but I would rather
have a quran verse) that says the state has to do this?
The Quran has verses that talks about how people should govern themselves. The Quran says that the prophet is not in charge of the affairs of Muslims.
Sahih International: But your people have denied it while it is the
truth. Say, "I am not over you a manager."
Yusuf Ali: But thy people reject this, though it is the truth. Say:
"Not mine is the responsibility for arranging your affairs;
Shakir: And your people call it a lie and it is the very truth. Say: I
am not placed in charge of you. (6:66)
So, it's upto Muslims to decide what type of laws they will follow which is prescribes in the following verse.
Sahih International: And those who have responded to their lord and
established prayer and whose affair is [determined by] consultation
among themselves, and from what We have provided them, they spend.
Yusuf Ali: Those who hearken to their Lord, and establish regular
Prayer; who (conduct) their affairs by mutual Consultation; who spend
out of what We bestow on them for Sustenance;
Shakir: And those who respond to their Lord and keep up prayer, and
their rule is to take counsel among themselves, and who spend out of
what We have given them. (42:38)