Let's say I have the wheat/pumpkin and pumpkin/turnip regular customers in my play area. I play the Merchant ("You may buy, for 1 cash, 1 of any vegetable that you must deliver at least twice to your Regular customers in this round.") and buy 1 pumpkin for 1 cash.

Later in my same turn the wheat/cabbage customer is revealed (it was the bottom card of a two-pack). Am I able to then buy a wheat for 1 cash?

In other words, does the Merchant stay active for the rest of my turn?

Nope. When you bought the pumpkin, the Merchant went on the discard pile.

Yup.

All helpers are placed immediately on the discard pile before you actually do the action. This is important, for the timing when the Foreman can be played.

It sounds like David was keeping played cards in his play area, then discarding them later.

This answer is almost right (the Merchant has to be discarded immediately after use, yes), but there are exceptions:
Eg the Maid stays active for the whole phase. But in this cases this is mentioned explicitly in the text.

While we're on the subject, I have a similar (but different) question WRT the Saleswoman. Option 2 on this card states that "During the current Action phase, casual customers pay a bonus of +2". I understand the effect remains for the rest of the Action phase. But in this case, when does the Saleswoman get discarded? Immediately, or at the end of the Action phase? In a game we played the other night, we ruled that it could be discarded immediately, while allowing the effect to remain active. This reduced the cost of a two-pack for the player who did it. I'm not sure whether that was the correct call or not.