I agree with your assessment. Either add an indefinite article or get rid of the preposition.

There's no real difference in meaning.

It's possible that "to be of some help" implies optimism (that you will likely be helpful) while "to be of any help" implies the possibility that maybe you will not be helpful at all.

It's more likely that speakers will use these without any sense of this distinction though.

This doesn't seem like much of an error because the rules have become so relaxed, but traditionally the relative clause "that" is used with restrictive relative clauses: those that are necessary to identify a noun and are not set off by commas. The pronoun "which," on the other hand, is used with nonrestrictive clauses, which are optional, add extraneous information, and are set off by commas.

...the Buddha, which has nine figures, made the religion... (extra information about the Buddha)
...the Buddha that has nine figures made the religion...
(necessary information to identify the Buddha)

I would use "had fallen." Because the simple past verb "said" refers to an action that occurred in the past, it would make sense to use the past perfect "had fallen" to indicate that the occurence of his birthday predated John's statement.

Make that "grammatically fine."

I think the sentence is grammatical fine but informal and more likely to be spoken than written because of its pronoun reference.

Traditionally, a "dependency" was a component that relies on something else, like a commonwealth that depends on a country for defense, while "dependence" was the state of being dependent. I'm not sure when the meanings merged, but it seems like the words are now used interchangeably.

Having said that, "dependency" sort of makes me cringe. It sounds like Stephen Colbert's word "truthiness." I think if you look up the word "dependency" in the dictionary, the definition ought to be "reliantness."