have + obj + to + V

I'm not a linguist, but this is my take (American English):

pash said:

Meet you at six. Just now I have a letter to write.

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Obligation. This could also be worded: I must write a letter. Possession is somewhat implied since no one else could write my letters.

pash said:

I'll be with you later. I have a famished dog to feed.

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Again, obligation. Note that "famished dog" ought to be qualified with "a" since there's only one dog. "I have a dog to feed" implies that I am obligated to feed the dog, but there's no proof that I own the dog in question. I could be talking about my dog, but I could also be talking about my friend's dog.

This is essentially using the kind of parsing for modal constructions that associates the modal verb/helping verb with one of six functions.

Technically it is all three, but the primary function is obligation/compulsion.

"I have a letter to write" and "I have to write a letter" are the same. Both imply that I must write the letter. I might or might not want to do it, but I am obliged to write the letter.

"I want to write a letter" implies willingness to write the letter but no sense of obligation. I want to do it, but without other information there's nothing to say that I have to do it.

Pash, I'm not quite sure what you mean by possession.

pash said:

<Technically it is all three, but the primary function is obligation/compulsion.>

What do you mean by "technically"?

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It's also possession because one can neither write a letter one does not have, nor feed a dog that one does not, in some sense "have."

Look at it this way; when you take a friend to the airport you say, "Look, there's your plane."

Mind, the plane is owned by Delta, but it's the one your friend is going to board.

All three; it's the speaker's particular man. Not just any man.