Is phrase correct?

I think No. 2 is correct.

"This" does indicate the robbery, and "that" would be used to refer to the robbery in past tense (not immediately past) or if someone else had done the robbery.

For example, if someone is banging on the floo you'd say, "Don't do that!"

But if you're banging on the floor, you'd say, "Don't do this" (as in, what I am doing).

What about the second question?

If I were a teacher, I would accept either answer. I think 'this' refers to the robbery. On the other hand, the rule in reported speech is to change 'this' to 'that'.

Is the phrase in bold correct? If it isn't, how should it be phrased?

I would need another example to consider changing this to that... I've never heard of that rule!

Look up the definitions of the two words...

If you're eating something now, you'd say, This is good!

If you ate something a while ago, you'd say, That was good.

If your friend is eating something, you'd say, That looks good.

This is for YOU, that is for SOMEONE ELSE, and for past tense.

I have no idea about the this-that rule.

Again, look up the definitions of the two words.

If I were a teacher, I would accept either answer. I think 'this' refers to the robbery. On the other hand, the rule in reported speech is to change 'this' to 'that'.

Is the phrase in bold correct? If it isn't, how should it be phrased?

You've misunderstood the question. I'm asking whether the part in bold is correct grammatically.

Sounds okay to me.

You guys have rules that no one who actually speaks English has ever heard of. This is one. In your example of direct speech, there is no stated subject to which "this" refers--it is only clear by implication what is meant--if you've got robbers and they're (apparently) in the act of robbing someone, which is illegal, you can infer that "this" means the robbery from your general knowledge of the context. However when you turn it into reported speech, you lose that context and that immediacy, and it's not immediately obvious what "that" refers to. Therefore only "report the robbery" really works, because that provides the context again.

I can't see how your rule applies. John said, We're going into this house", does not necessarily translate to John said they were going into that house, OR John said tha they were going into this house. "That" and "this" are in relationship to the speaker of the reported speech, not necessarily to the original speech. I don't think you can make a hard and fast rule like the one you're positing.

The phrase "the rule in reported speech" is correct grammatically, however I don't think the rule itself is correct--or else it is too simplistic to actually conform to actual usage or meaning.