Is it allowed to work as student (intern) in Luxemburg given that I hold a valid residence permit and a valid working permit in Germany?
Is it also allowed for those who have Blue-Card issued from Germany?
Working permits are still national matters in the EU and your German working permit does not allow you to work in Luxembourg. To the extent that your internship counts as work, you would therefore need some form of authorization from Luxembourg.
For less than 90 days, you can however travel to and stay in Luxembourg on the basis of your German residence permit and would only need a permit to work. For more than 90 days, even staying without working would be forbidden and you would need a Luxembourg residence permit/visa anyway.
A German blue card is a type of residence permit and as such also allows the holder to stay in another Schengen country under the conditions I just described. On top of that, there are special rules to make it easier to take up employment in another participating country but no unconditional right to work without applying for a local blue card.