"This" is analogous to "here", whereas "that" is analogous to "there". If you mentally substitute this/that with "this here" or "that there", one will often seem to be the obvious choice. Another way of thinking about it is if you imagine actually showing the other person the thing you are talking about, would you point to it, or hold it up to show them? If you would point to it, use "that"; if you would hold it up to show them, use "this".
For a concept or idea introduced a few sentences before, I think the key thing is whether it was introduced by you or the person you're talking to. If you introduced it, you're "holding" the idea - use "this". If they introduced it, they're "holding" it - use "that".
As for your last sentence ("Is there a rule describing proper usage of these words in cases like this?") "this" sounds much more natural and correct. I think that's because you are presenting some examples for consideration, so you can imagine that you are holding them in your hand/head and showing them to the target of your question, hence "this".
Interestingly, if I were to then refer to your examples I could use either "this" or "that". Both would sound fine, but have different implications. "This" implies that I have taken your examples/concept/idea into my own headspace to examine and consider close-up, and am still considering them. "That" implies that they are with you, rather than me; either because I'm considering and commenting on them from "over here", as your examples, or possibly that I've brought them into my own headspace, considered them, and have given them back to you prior to commenting on them. Either way, the implication is that they are your examples - I haven't taken on shared custody of them.
It depends on what you mean by purpose:
something that one hopes or intends to accomplish => "to this purpose" (as in "to this end", "to this intent", "to this objective", "to this target", ...)
the action for which a person or thing is specially fitted => "for this purpose" (as in "for this function", "for this job", "for this position", "for this task", ...)
For example, you will find both meanings in this gide "The Structure, Format, Content, and Style of a Journal-Style Scientific Paper":
In scientific studies, the use of this word (significant) implies that a statistical test was employed to make a decision about the data; [...].
Limit the use of the word "significant" to this purpose only.
Organize your presentation so your reader will understand the logical flow of the experiment(s); subheadings work well for this purpose
Best Answer
"Out for lunch" makes me think the person will be bringing the food back with them.
I have heard the other variations and they seem to mean various things:
I have also heard the phrase used to imply someone is "out to lunch" or not entirely there mentally. Context seems to be the only clue that this meaning was intended:
When talking to someone while eating, I find this more common: