I would like to use the quiz for my homepage. But I would like to know, if I can set the access to the quiz just for users that did not pass the quiz yet. All members that did pass the quiz already should not be allowed to review the questions of the quiz. Is this possible or will I need custom development?

Yes, this is possible. You should set Attempt Count to 1 in Additional Settings and Access to 'Registered' in quiz settings. This will allow logged in users to pass the quiz only once.

that is not really what i want, i would like to allow them 3 attempts, but once they solve the quiz they should not be allowed to view the questions again.

If you set up the settings we have mentioned above user will be allowed to pass the quiz only once. Once he has finished the quiz he will not be allowed to pass it again. But if he started passing the quiz but NOT finished it (for example somewhere in the middle he closed the browser) he will be allowed to go back and continue from the last question he has seen.

Looks like this behavior satisfies your requirements.