Aside from a couple nitpicks, your sentences are grammatical, and would easily be understood by a native speaker. The third sentence would more likely be
After his wife died, he fell apart.
to make the verb tenses match, unless there's a good reason to use the perfect tense. I wouldn't use "now" in your second example, and it would be more natural to combine the two sentences:
I have used my dictionary so much that it has fallen apart. (formal)
I've used my dictionary so much that it's fallen apart. (casual)
I suspect that there are two reasons why the perfect tenses of "to fall apart" would be less common. The first reason is that completely falling apart is usually a transient state. Once the falling apart is over, you'd be more likely to say that something "fell apart" or "finally fell apart". The second reason is that people tend to talk more about problems that are still happening. For those cases, "is falling apart" or "has been falling apart" are more common.
Let's look at your examples again.
My poor old boots fell/have fallen apart.
With "fell", this is a simple factual statement about your boots. With "have fallen", it sounds more like the answer to a question. "Do you want to go for a walk in the snow?" "I'd like to, but my poor old boots have fallen apart." If the listener knew that the boots were falling apart, you'd probably say that they "finally fell apart".
My dictionary has fell/has fallen apart. I have used it so much.
Usually you don't use a book until it completely falls apart. If the book does fall apart, it's not really an ongoing state. The book fell apart. That's it. Saying "has fallen" here sounds a tiny bit melodramatic.
San Diego's public buildings are falling/have fallen apart, but the city refuses to do anything about it.
As with the dictionary, buildings rarely collapse completely. They just gradually get worse over time. Saying that the buildings "have fallen" apart implies that they're (almost) unusable.
She liked her old apartment, but the neighborhood **was falling/had fallen* apart.
As with the building, a neighborhood's decline is usually a lengthy, continuous process. Saying that the neighborhood "had fallen" apart implies that it reached a terminal state of badness.
My conclusion from all of this is that "has fallen apart" describes a total collapse that is also an ongoing state, which is just not that common. A notable exception would be:
My life has fallen apart.