Interesting question! Make and Let have subtly different meanings in certain cases, and are not interchangeable at all in others. To look at your examples:
Don't make me feel alone.
Make here implies that the person you're talking to is intentionally trying to cause you to feel alone. They are making you feel this way, and by stopping their behavior you wouldn't feel alone anymore.
Changing this to let, on the other hand:
Don't let me feel alone.
Let here means that you already do feel alone, and are asking someone to help make you not feel alone. In this case their action will stop you from feeling alone, and if they do nothing you will continue to feel alone.
So here the small difference between make and let actually completely changes the meaning of the sentence.
Make me understand what you mean to say.
This is a command; you want to understand what they're saying, and you insist that they make it so you understand. This is perfectly acceptable, though a bit authoritarian; it's more likely one would say "Help me understand what you mean to say" as a request than to pose this command.
That aside, let is not at all appropriate here.
Let me understand what you mean to say.
This doesn't really make sense; asking someone to let you understand implies that they are currently preventing you from understanding and you would like them to stop. This seems an unlikely circumstance, and probably doesn't convey the meaning you're searching for.
Now for your last case:
You are requested not to let me feel alone.
This sounds oddly stilted and formal, but is grammatically correct. It carries the same meaning as "Don't let me feel alone", except it is phrased as a request rather than a command. If you're trying to politely ask someone to keep you company, for example, you might try saying:
Please don't let me feel alone.
Or if you specifically don't want the person to leave because this is the action that will make you feel alone, you'd most commonly say:
Please don't leave me alone.