Learn English – how and when to use a preposition for the verb ‘ make it ‘

There are many different phenomena in the sentences you list.
Most of them don't really have much to do with prepositions in general, but with specific constructions.

1. I let you do it is a complex sentence with an unmarked infinitive complement.

2. I allow you to do it is a complex sentence with a marked infinitive complement.

3. I said X to you a week ago is a simple sentence with a direct object X and an indirect object you. The verb say does not allow the Dative Alternation.

4. I told you X a week ago is the same structure, with the same objects.
   But the verb tell does allow the Dative Alternation, and this is the alternate version.
   The unalternated version, I told X to you a week ago, is also correct.

The "mark" in both cases 1 and 2 is the infinitive complementizer to.
This is not a preposition and does not behave like one.
For one thing, certain verbs (like let) allow one to delete to before infinitive complements,
while most verbs, like allow, require a to to mark infinitives.

In cases 3 and 4, the to is part of a bitransitive (transfer) verb -- one that has two objects.
The indirect object in such a sentence can usually be marked with to, if it follows the direct object.
However, if it precedes the direct object (the Dative Alternation), then to is not allowed.

You actually have the question backward! :) ... you should be asking When does an Indirect Object require a preposition?

(For those unfamiliar with the term, an Indirect Object is a secondary object of a transitive verb: it designates the entity which receives the Direct Object or for whose benefit the action is performed on the Direct Object.)

Subject - Verb - Indirect Object - Direct Object

This construction is the standard order in all di-transitive situations—those in which the verb takes both an Indirect Object and a Direct Object.

I told the police my story. OR I told them my story. I gave my son a bicycle. OR I gave him a bicycle.
I built my niece a treehouse. OR I built her a treehouse.

Sometimes, however, the Indirect Object has to fall after the Direct Object. When that happens, the syntactic relationships are no longer clear, so the Indirect Object is ‘marked’ with a preposition. There are two circumstances when this happens:

• When the Direct Object is a single pronoun it is too ‘light’ for its role to be clear, so it is moved immediately after the verb:

  ^∗ I told the police it. → ^okI told it to the police.
  ^∗ I gave my son it. → ^okI gave it to my son.
  ^∗ I built my niece it. → ^okI built it for my niece.

• When the Indirect Object is ‘heavy’, more than a couple-three words long, it draws focus for too long and it makes it difficult to recognize the Direct object, so it is moved after the Direct Object:

  ^∗ I told the police who came after I called 911 my story.→ ^okI told my story to the police who came after I called 911.
  ^∗ I gave Richard, my ten-year-old son going to school just a few blocks away, a bicycle. → ^okI gave a bicycle to Richard, my ten-year-old son going to school just a few blocks away.
  ^∗ I built my niece, my sister's daughter Lily, a treehouse. → ^okI built a treehouse for my niece, my sister's daughter Lily.

  These ‘heavy’ Indirect Objects may not seem all that hard to process; but that is because you are encountering them in writing, where you have the advantages punctuation to clarify structure and of seeing the entire sentence at once. In speech, where your hearer must process your sentence as it emerges, it isn’t so easy, and these constructions are driven by speech patterns.

Two general “rules” are operating here.

• The first is that we like to keep the core constituents of a sentence—the Subject, Verb, Indirect Object and Direct Object—as close together as possible. That drives ‘heavy’ constituents towards the right end of the sentence, where their length provides the least interruption.

• The second is that we like to put the ‘new’ information in a sentence at the right end. That drives pronouns towards the left, because pronouns are almost always ‘old’ information—they usually refer to entities which have already been introduced.