When the woman answers Jesus, "Yes, Lord," is she agreeing or disagreeing? Most translations use an adversative like "but" for the following γάρ even though that gloss does not appear in the lexicon. Are they right to do so?

The Greek word ὁ is much more than what we call the “definite article.” Depending on context, it can function as a demonstrative or a possessive pronoun. In this verse, it is called an “anaphoric” article.

In English, modifiers need to be next to the words they modify. In Greek, they don't care because they have a different system of linking modifiers to their head noun. So what's a translator to do?

What should a translator do when the actual words don't make sense? Do we let the reader figure out the meaning, or do we help?

As you get into Greek, you will find that the grammar is not always correct. Sometimes, like in Revelation, the grammar can be explained by the fact that John is in an ecstatic state and the grammar irregularities enhance the message. But even in non-apocalyptic literature, you will find that the grammar rules are not always followed. This should not affect your view of inspiration. All good writers violate grammar, periodically, for effect, or for other reasons.