To begin with, look at your sentence:

"Members of the 16th Lok Sabha worked harder in the budget session than they have in the last 10 years, parliamentary data shows.

I recommend looking at it in two parts:

1. Members of the 16th Lok Sabha worked harder in the budget session
2. than they have in the last 10 years, parliamentary data shows.

Sentence 1 tells you that members of the 16th Lok Sabha worked harder in the budget session. The "budget session" refers to the present budget session, that happened just now, or during the 16th Lok Sabha, if you will.

Sentence 2 introduces a comparison- with how much they have worked in the last 10 years. Now, the idea behind using "have" is to introduce a continuum, a matter of fact that "has been happening" regularly over the past 10 years.

If you use "had" instead, it gives the idea that something HAS happened at a certain point in time, and then that's it. It didn't happen anymore. For instance:

They studied harder this term than they had last year.

Notice how "had" is used to indicate what they did just once, that is last year.

Again, if you now use "have" in the same sentence, see how you get a different meaning:

They studied harder this term than they have in the last five years.

Here, "have" tells you about something that hasn't occurred just once, but has been occurring regularly over a course of time, that is five years.

It's interesting to note that using "had" in part 2 of your original sentence wouldn't make it incorrect, technically. There is a very fine line between using "have" and "had" to denote a continous action in the past tense. As I explained with my previous example, if you use "had" in sentence 2, it indicates that the members of the Lok Sabha had worked up until now, which doesn't fit in well with sentence 1.

And lastly, do take a look at this question on the EL&U site: How do the tenses and aspects in English correspond temporally to one another?. It should give you a clear picture about everything.

Your daughter is correct: in standard British (or US) English, it should be “Yes, they do.”

The key here is that do, not have, is the auxiliary verb. Have can sometimes be an auxiliary, but in this sentence it’s the main verb. So:

“Do they like pizza?” “Yes, they do.”

“Have they had lunch yet?” “Yes, they have.”

“Do they have some?” “Yes, they do.”

(In some dialects, your teacher’s form would be fine; also, it was quite standard historically. But I think (though I’m not sure) that most speakers who’d use that form would also pose the question differently, inverting the main verb without using an auxiliary do, and would still match the verb of the answer to the initial verb of the question: “Have they any?” “Yes, they have.” parallel to “Heard ye the tidings?” “Yes, I heard.”)