Learn English – Is “wasting both of our times” correct

"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.

The phrase "in 15 minutes" is correct. It means after 15 minutes elapses, which is precisely when the show starts. The problem with "15 minutes later" is that it's not clear that it's relative to the present.

One show can start 15 minutes later than another, but if a show simply "starts 15 minutes later" -- later than what? The phrase "later" has to compare things such that one of them can be later than the other(s).