「いただきます」 to accept to a food offer?

Question

In this JLPT N3 audio practice question (#3), the correct reply to the statement is 3.

コーヒー、もう一杯いかがですか。

1. けっこうだと思います。
2. おかげさまで。
3. いただきます。

If the question translates as "How about another cup of coffee?" then I'm confused why 3 is correct. First, I thought 「いただきます」 as a set phrase only applies just before a person is about to put food to mouth; second, in this setting that I assume to be a customer and server, I'm confused why the customer would be speaking keigo. Obviously there's some detail I'm missing here.

Also, why is answer 1 incorrect? If 「結構です」is an acceptable way to turn down an offer, why would a slightly less confident 「結構だと思います」 be out of place? Is 「結構です」strictly a set phrase that can't be modified?

Answer 1

Here, いただきます is just "I (humbly) receive/take/accept it", a humble version of もらいます and a polite version of いただこう as well. いただきます as a set phrase is of course related to this, but here it's used in a more literal sense.

It's not very common to use keigo to a waiter at cheap izakaya, but it's fine and natural at classy restaurants. And an exchange like this can happen in a situation where someone has invited someone to their home, too.

Is 「結構です」strictly a set phrase that can't be modified?

Sort of. 結構, いい, よい, だいじょうぶ and so on have seemingly contradictory meanings, so you should not change its form lightly. 結構だと思います can mean "Your idea sounds good" as well, so it's confusing. To turn down this offer, you have to say 結構です, 大丈夫です or いいです (or 結構/大丈夫だ if you don't mind sounding blunt and arrogant). Related: How to "shoo away" a sales clerk? / How to respond to ポイントカードが宜しいですか。