Speaking as someone who went to a lutheran seminary, let's attack these questions one at a time, shall we?
Is this "Law" of ("Law and Gospel") referring to the Old Testament law?
The Solid Declaration of the Formula of Concord says in Section V "Law and Gospel"...
Anything that preaches concerning our sins and God's wrath, let it be
done how or when it will, that is all a preaching of the Law. Again,
the Gospel is such a preaching as shows and gives nothing else than
grace and forgiveness in Christ, although it is true and right that
the apostles and preachers of the Gospel (as Christ Himself also did)
confirm the preaching of the Law, and begin it with those who do not
yet acknowledge their sins nor are terrified at [by the sense of]
God's wrath; as He says, John 16:8: 13] "The Holy Ghost will reprove
the world of sin because they believe not on Me." Yea, what more
forcible, more terrible declaration and preaching of God's wrath
against sin is there than just the suffering and death of Christ, His
Son? But as long as all this preaches God's wrath and terrifies men,
it is not yet the preaching of the Gospel nor Christ's own preaching,
but that of Moses and the Law against the impenitent. For the Gospel
and Christ were never ordained and given for the purpose of
terrifying and condemning, but of comforting and cheering those who
are terrified and timid.
For Luther, the "Law" was anything that stung the conscience of its own sins against GOD and neighbor, while "Gospel" or "Grace" was anything that communicated the love and forgiveness of GOD.
If so, do Lutherans believe that these Old Testament laws should be enforced as equally as the rules set forth in the New Testament?
Lutherans believe that not all Old Testament laws should be enforced equally, but that there is a timeless truth behind each of the laws that can be applied to today's context. For further information on this please see...
The Epitome of the Formula of Concord section VI entitled "The Third Use of the Law"...
VI. The Third Use of the Law.
1] Since the Law was given to men for three reasons: first, that
thereby outward discipline might be maintained against wild,
disobedient men [and that wild and intractable men might be
restrained, as though by certain bars]; secondly, that men thereby
may be led to the knowledge of their sins; thirdly, that after they
are regenerate and [much of] the flesh notwithstanding cleaves to
them, they might on this account have a fixed rule according to which
they are to regulate and direct their whole life, a dissension has
occurred between some few theologians concerning the third use of the
Law, namely, whether it is to be urged or not upon regenerate
So, for Lutherans the Law has three purposes, 1) restrain evil 2) show humanity its evil and 3) as a guidebook for how Christians ought to live in grace.
I hope this helps!