Paul?s gospel, summarized in Romans 1:16-17, is simply this: God alone
provides salvation (righteousness) as a pure gift for those who accept
it (those who believe). If salvation is a gift, then what is the
purpose of the law? In other words, what is the relationship between
faith in Jesus Christ and the moral guidelines Paul enumerates in
chapters 12-15? According to Paul, why is this the case?

Maybe the question lies in what you define salvation as. If you define
it as righteousness, then you have to ask yourself what do you mean by
righteousness?

If righteousness means obedience to the law, then there you have your
reason for the law...to define righteousness and thus salvation as
obedience to law. Thus Paul's gift of righteousness would be in
essence the law or in other words the way  of salvation.

As if that wouldn't confuse anyone further...If Paul were still around
he'd even be confused!

"The golden rule is endorsed by all the great world religions; Jesus,
Hillel, and Confucius used it to summarize their ethical teachings.
And for many centuries the idea has been influential among people of
very diverse cultures. These facts suggest that the golden rule may be
an important moral truth."
http://www.jcu.edu/philosophy/gensler/goldrule.htm

The above is one man's statement, but I don't think we need to argue with it.
If God alone provides salvation to those who believe, for Christians
that means also believing in the tenets Christ taught (and not only
he).
To make a sweeping statement: all law (well, maybe not tax law  ;-)  )
is a codification of the golden rule.  If we all adhered to it, we
wouldn't need all the laws that spell out in ever increasing detail
what we should and should not do: be it murder, discrimination, not
planting trees where they could bother the neighbor, driving
recklessly ...
I think it was Thmas Jefferson who said something to the effect that
laws are there for those who would break them, not for those who would
not.

So by obeying the law, we are only abiding by the golden rule, and from my 
glance at Romans 12-15, what Paul enumerates supports this.

So not to get philosophically deep, but it really comes down to what
salvation is defined as.

You can take it two ways, either salvation is a (1) destination or a
goal or (2) a state of being

If it's a goal, then "gift of salvation" could then essentially be the
method to get there. It's as if Paul says, "here's a map to salvation,
follow it and you have your gift (Christ)" So in essence, the gift of
salvation is the law.

If it's a state of being, such as righteousness, then it's still the
same thing, the method.

One would tend to think though as salvation as a destination because
if it's a state of being, then there are many who are currently in
"salvation" as opposed to being headed in that direction.

So not to get philosophically deep, but it really comes down to what
salvation is defined as.

You can take it two ways, either salvation is a (1) destination or a
goal or (2) a state of being. If it's a goal, then "gift of salvation"
could then essentially be the method to get there. It's as if Paul
says, "here's a map to salvation (Christ), follow it and you have your
gift" So in essence, the gift of salvation is the law.

If it's a state of being, such as righteousness, then it's still the
same thing, the method.

One would tend to think though as salvation as a destination because
if it's a state of being, then there are many who are currently in
"salvation" as opposed to being on the path of salvation.

It is a teaching tool to show up sin.  This is evident in Romans 7:7
and in Galatians 3 which covers your question.

The Law also provides a basis for God to judge evildoers and to
approve the obedient (even if the Law as such is not their motive).

the purpose of the law is to keep us out of trouble.. if you eat fatty
pork all day you will get fat and unhealthy. is it a sin against your
soul? no. its just against your body.

jesus came, and he said something to the effect of "i did not come
here to change one bit of the law, i came to fulfill it".. its awesome
because now all the blood sacrifices and lambs and stuff, we no longer
have to do.. because jesus became our spotless lamb.

some people think the law is old testament and non exsistant.. i guess
they can go murder people and break all of the 10 commandments as
well. but if pork made people's bodies unhealthy back then, it will
now too.

now, there are also verses in the NT about if you go to someones house
and they feed you pork, dont worry about it.. accept their generosity
for cooking you a meal and eat it with delight.. everything you touch
is blessed. dont worry about simple stuff like that though. you have
jesus inside you and he has overcome the world for you. the holy
spirit will guide you.

this example was just pork because it just popped into my head. 

but jesus knew we could not keep every rule and every comandment of
the law.. if you break one it is as if you broke them all.. so we are
all murderers, all adulterers.. we cant do it on our own.. so we have
to fall back into the arms of grace.

personally i dont think jesus wants us each day to say "ok i cant do
this this this" but rather we should just love him with all our heart
and all all the small detailed, nit picky things of the "law" will
fade and jesus will be magnified when our focus is on him.. my point
is, dont get bogged down.