Yes, this is all correct. I'm not sure how natural it sounds, because "furnish" (in this sense) is quite uncommon, mostly used in legal or technical documents. A more common alternative would be "provide". But all of your examples are grammatically correct.
"Details" is the direct object of the verb "furnish". "Me" is the indirect object. Unlike some languages, these types of object are not marked on the words themselves, but by the way the sentence is built around them.
Unfortunately for learners, not all structures are used with all verbs. I have marked inappropriate uses like so:
don't say it this way They aren't necessarily wrong, and in fact another speaker (from another country, perhaps) might say it that way, but to my mind they don't sound natural.
[VERB] [DIRECT] to [INDIRECT]:
She was able to furnish the details of the incident to me.
She supplied information to me.
She gave a gift to me.
She wrote a letter to me.
[VERB] [INDIRECT] [DIRECT]:
She was able to furnish me details of the incident.
She supplied me information.
She gave me a gift.
She wrote me a letter.
[VERB] [INDIRECT] with [DIRECT]
She was able to furnish me with details of the incident.
She supplied me with information.
She gave me with a gift.
She wrote me with a letter.
While double-checking my answer, I found a source that says "with" makes a third category, the "prepositional object". I'm not familiar with that; to me, it's always just been direct and indirect. However, this might be useful to explain why this form is not used in so many cases.
[VERB] [DIRECT] for [INDIRECT]:
She was able to furnish the details of the incident for me.
She supplied information for me.
She gave a gift for me.
She wrote a letter for me.