Still flutters there, the sole unquiet thing. Methinks, its motion in this hush of nature Gives it dim sympathies with me who live,
Coleridge jumps to a new stanza before he finishes this thought It indicates that his mind is going somewhere else, from thinking about how quiet it is to a new line of discussion.
The film is the only thing that isn't quiet, thanks to its persistent fluttering. This is part of the reason it drags Coleridge's mind away from the emptiness of thought (the "vexe[d] meditation" of the earlier lines), which was provoked by total silence, into some of the "abstruser musings" he's trying to have.
Since the flame is the only thing that isn't quiet, and he's the only person who's awake and intentionally thinking, he sees that it has "dim sympathies" with him—it's vaguely similar.
There's assonance with the short I vowel sound here, too: "Gives it dim sympathies." Hit up "Sound Check" for more.
Making it a companionable form, Whose puny flaps and freaks the idling Spirit By its own moods interprets, every where Echo or mirror seeking of itself, And makes a toy of Thought.
The moving film seems like a "companionable form" for the same reasons stated in the discussion of the last lines—Coleridge is the only person awake, and the film is the only thing moving.
He starts to see his own thoughts and mind as being similar to the film. (Is it us, or is Coleridge really getting into this pile of dust in his fireplace?) The "idling Spirit" is at rest, watching the mind move (since "the Spirit" is the same as consciousness or awareness, which simply beholds what thoughts are doing). In the same way, the fire in the fireplace is at rest, metaphorically watching the film move.
The Spirit—according to Coleridge—sees a mirror image of itself in the mind and in the rest of earthly reality. Thoughts are a toy-like imitation of that Spirit, created by it for its own amusement—deep stuff.
Form-wise, there's alliteration here with "flaps and freaks" (see "Sound Check" for more). A "freak," in this case, is just a wild, unpredictable movement.