Should we start re-posting questions from SO which are tagged as licensing?
Ask if you think that the question is relevant for this site and you care about the answer.
If you reuse material from elsewhere, make sure to provide suitable attribution. Do link to the original so that people can browse the answers that are available there.
Make sure that you ask in a form that's suitable for this site's audience. For example, unlike Stack Overflow, we don't automatically assume that a question is about software. Conversely, we automatically assume that a question is about open source.
If we are to do so, shouldn't that be task for mods so the site does not get flooded with duplicating/low quality questions (made wikis maybe?)?
Don't ask low-quality questions! Ask only good questions, whether you found them elsewhere or thought them up by yourself.
Wiki is irrelevant. We don't really do wiki questions anyway, only wiki answers.
If such task should be carried from the community as a whole, should the questions be left open or quality/accepted answers from the originals should be reposted as well?
Repost an answer if you think that it's a good one and relevant to our community. Provide suitable attribution, of course.
It would be better to produce better answers. Licenses, for example, are outside Stack Overflow's expertise, so answers can sometimes be uninformed. For example, if Stack Overflow has several answers that provide complementary material, then rather than copy all of the complementary answers, write a single answer that combines the material in one smooth flow.
Some users might exploit the opportunity to gain some rep (rep-hounding) in time and start posting duplicate questions and questions with low quality or which are not quite fitting the definition of this site
Don't make low-quality posts. That does regardless of whether you're copying or getting inspiration from elsewhere.