First, the topic of your question.
Do the extra tables in the Background section of the 5e PHB count as Characteristics?
Yes, but also note they are not 'extra' . As you have noticed, each background type has one of these that are not a bond, flaw, ideal, or trait; the heavy 4 that every background has. This other table is the primary characteristic of each background, be it why they went into seclusion, to what kind of criminal they are, and is not something extra or optional.
As a result,
Do these count as Characteristics, insofar as they interact with Inspiration mechanics?
Yes, they help give the player a more precise focus as to why they are their chosen background, or how their character OF the chosen background behaves. A Hermit who was exiled for a crime he didn't commit would have very different drives and behaviors than a Hermit who is the caretaker of an ancient rune/relic.
Which opens up potential roleplaying and story weaving ideas for the Dungeon Master to include in their campaign. Where's the relic? What is the relic? What's the Hermit's ties to the relic? If said Hermit does a good job playing to that background in character? Why not award inspiration.
Of note, when you ask about Inspiration mechanics, there are no defined inspiration mechanics in the book(s); they give a loose guideline and leave it up to the DM's to award inspiration as they see fit. So do you, as the Dungeon Master, feel like these better allow the players to interact with YOUR inspiration mechanic?