Hello! You speak English well.
In Orthodoxy, the Noahide Laws are not really emphasised, even though they were approved by the Council of Jerusalem (50 A.D.) and are still accepted by the Church. This is largely because the Laws are mostly natural law, especially considering how even the pagans had similar laws. So, we're still under obedience to them, but they're really common-sense morality.
Your second link is a little more interesting. He essentially says that we are still under obedience to the Law of Moses, leveraging his argument on how
the question [in Galatians 2:14-15] wasn't "Why are you over there trying to get Gentiles (Torah breakers) to live like Jews (Torah keepers)? Don't you know the law is abolished?" The real question was "How can you ever compel the Gentiles to keep the Torah (like Jews are supposed to) if you are a sinning yourself?" Both Greek texts are compatible with the latter understanding, but the Nestle-Aland text would contradict the common Christian interpretation of these verses.
Even if we accepted his translation, we still don't necessarily have to follow the Mosaic Law because Christ fulfilled the Law, consequently freeing us from having to observe it, because we could not keep it and would sin under it (The Menaion speaks of the old Law as a "curse."). St. Paul especially drives this point in other letters. As the hymns of the Church speak, Christ abolished the sacrifices by His sacrifice on the Cross, he abolished the Sabbath laws by His rest in the tomb on the Sabbath (the Holy Saturday service says "The King of the ages, having through His Passion fulfilled the plan of salvation keeps Sabbath in the tomb,
granting us a new Sabbath."), and abolished circumcision by His circumcision - as the Menaion says "The transcendent Word Who became incarnate hath been circumcised, that the law might cease
. . . and in accordance with the law didst fulfil the law
." All of Christ's life points to the abolition of the Mosaic Law. What is Christ to these judaizers? Unless His life and His Passion point toward a new covenant (which He established at the Last Supper in Luke 22:20), then Christ's mission is no longer salvific, and He only becomes a special prophet (much like in Islam). But this plainly contradicts Scripture, and the earliest traditions of our Church confirm that we don't have to obey the old law. Hopefully this helps a little, and hopefully I didn't appear too irate - this subject just bugs me.