> -Góðan daginn!
> -Góðan dag!
>
> or
>
> -Góðan dag!
> -Góðan daginn!
>
> 1. If the first person to speak picks one of the pair,
> must you pick the other? All the speakers on my tape
> seem to do so.

From my point of view the other person almost always says
'Góðan daginn!' and I almost always say 'Góðan dag!' -
which fits with your theory. I don't know how general
this rule is - but it fits with the ding-dong theory ;)

> 2. Obviously, the case involved here is accusative (masc. sing.).
> Góðan daginn, according to strict rules which I can see may not
> apply here, is surely incorrect grammar, as the adjective should
> be weak before a noun + article, so one would expect Góða daginn.
> Or is there another rule which makes it an exception?

And that's why I, being a pedant, always say 'Góðan dag!'. ;-)
I don't know how 'Góðan daginn!' developed; perhaps as some
sort of "dong" to the 'Góðan dag!' "ding".

- - -

In Icelandic the accusative seems to function as a sort of
"imperative" for the nouns. When we say things like:

Hnífinn, takk! - The knife, please!
Hníf! - The knife!

the accusative is used.

I might run for president with slogans like this one:

"Hauk Þorgeirsson sem forseta!"

In each case we can think of a verb that is omitted.

[Réttu mér] hnífinn.
[Við viljum] Hauk Þorgeirsson sem forseta.
[Guð gefi þér] góðan dag.

- - -

Icelandic has also developed a kind of vocative;
produced by using the definite form of the adjective
and the indefinite form of the noun.

Litli karl!
Góða mús!

Kveðja,
Haukur