I think you may be overthinking this a bit..
When using reported speech (but not a literal quotation), in general, you should describe the speech based on what is true relative to the current point in time (the point in time when it's being reported, not the point in time when it was spoken). So if the person speaking was talking about something that actually happened (or was to happen) in the past, then you should use a past tense for the reported speech. If they were talking about something that is still true now, or is still expected to happen in the future, then you should use present or future tenses.
Basically, if you can take off the "He said" and it still makes sense, then it's probably right.
He said he likes you. (--> He likes you.)
This says that (he said) he liked you when he said it, but it's also something that, as far as you know, is still true now, in the present (he also still likes you).
He said he liked you. (--> He liked you.)
This says that he liked you in the past, but he might not feel that way anymore.
He said he's finished studying. (--> He's finished studying.)
This implies that he's still in the state of having "finished studying" now. You might use this if it was something he just said a moment ago, so it still applies now. It would probably seem strange if he'd said it some time ago, though, because usually states like that don't last for a long time.
He said he'd finished studying. (--> He had finished studying.)
He was in the state of having finished studying then, when he said it. It doesn't necessarily say anything about what state he's in now (when you're reporting it).
He said he will call me the next day, but he didn't! (--> He will call me the next day, but he didn't!)
I think when you remove the "He said" it might be more clear why this sounds strange. You are saying that, according to what he said, he will call you some time in the future (from now), but you're also saying that instead he didn't (in the past), which doesn't make any sense. Assuming the "next day" you're talking about was some point in the past, then the action of calling is something that (should have) happened in the past (relative to now), so it should be past tense in reported speech, the same way it would be in non-reported-speech:
He said he would call me the next day, but he didn't! (--> He would call me the next day, but he didn't!)
This is actually correct, but admittedly we don't usually use "would" like this in a plain statement, so it sounds a little odd without the "He said". Another way to think of this is with something like "was going to" instead of "would", though:
He said he was going to call me the next day, but he didn't! (--> He was going to call me the next day, but he didn't!)
