Literacy, pens, paper, the printing press.
A written culture has different restrictions than an oral culture dependant on ease of repetition from memory.
According to the University of Wisconsin Digital Collections Center:
Beowulf is the oldest narrative poem in the English language, embodying historical traditions that go back to actual events and personages in fifth- and sixth-century Scandinavia. During the long preliterate centuries when these traditions were transmitted in the form of oral poetry, they were combined with with a number of legendary and folktale elements (among these are Grendel and his mother, the dragon, and probably the hero Beowulf himself). The written text of the poem, as we have it today, took shape in England during the middle or late Anglo-Saxon period and survives in a single manuscript from around the year 1000.
An oral tradition requires stories to be easily memorised and stand repetition many, many times, and passed on to the next storyteller. A strong metre and fixed structure with helps, along with alliteration (also found in Beowulf) and isn't unique to English. For example, the Finnish national epic, the Kalevala, is based on sung oral tradition and has a fixed pattern of stresses, with much alliteration and parallelism (repeating the previous line with different words but meaning the same thing). In fact, the hero, "steady old Väinämöinen" (an almost always alliterative "vaka vanha Väinämöinen"), is himself a storytelling wizard who plays a zither and uses his song-words for magic.
Once people can read and write, they no longer have a need for the storyteller to recite a story from memory, they can read it themselves, or have someone read it to them from a text. Over time, this gives rise to more creative ways of expression.
For more, here's a paper (PDF) on Oral Tradition & Its Decline by Indira Bagchi.
Different publishers are likely to handle the punctuation differently. I doubt that you'll be able to please them all, whichever convention you adopt.
At this point, I'll express my own preferences plus my reasons for those preferences.
Where you wrote
"We can't know for sure," he mused, leaning back in his chair, "Not until we ask more questions."
I would turn the dialogue into two separate sentences:
"We can't know for sure," he mused, leaning back in his chair. "Not until we ask more questions."
This is particularly justified here, because if you cut out the description and present the dialogue on its own,
"We can't know for sure, not until we ask more questions."
it becomes apparent that what you wrote is a run-on sentence (i.e. its clauses are grammatically entirely independent).
With your other query sentence,
I have to get out of here, he thought, But there's not enough time.
the interior monologue is not a run-on:
I have to get out of here, but there's not enough time.
Accordingly, I would put but in lower case:
I have to get out of here, he thought, but there's not enough time.
In that sentence, you don't need to capitalize but to signal the resumption of the monologue: the italics are already doing that.
Best Answer
Short answer: Yes.
“I care not” is idiomatic, but usually only as a standalone phrase. It has a slightly archaic feel to it, which makes it sound formal, which again makes it sound humorous. It is very common to use words or phrases from an inappropriate register to convey humour.
It is not uncommon to hear something like this:
Using the slightly archaic/formal pattern here, the speaker conveys a slight condescension toward all those people who dare to criticise his/her new couch, but also frames this slight condescension in a bit of humour so as not to be downright rude.
However, when care is transitive, this modal pattern is not common; the following is a lot more unusual:
It's not impossible, but it's even more highly marked, to the point that the speaker just ends up sounding odd, rather than using a hint of archaism and formality to add a nuance to the words spoken.