Like J.R., I don't see a good match between the language of the quoted sentence and any clear, recognized definition of churning.
The only definition of "churning away" that I'm familiar with carries a sense of purposeless activity; essentially it means "churning without a definite goal or foreseeable end," as in "He left the refrigerator door open, and the motor was just churning away all afternoon."
"Churning away" emphatically does not mean churning productively—not even at the impersonal, quota-driven level of "churning out," which, as Merriam-Webster's Eleventh Collegiate notes, involves "produc[ing] mechanically or copiously," but which frequently harbors an implication of poor or indifferent quality in the resulting output.
Making sense of the author's description of women churning away "dreams, fears, social [intrigues] and political intrigues" is complicated by the fact that one of the churned-away things (dreams) seems desirable, a second (fears) seems undesirable, and a third (intrigues) may be desirable or undesirable. Under the circumstances, it's hard to tell whether the women intend the churning to produce these things—or indeed whether the women have any clear intention at all with regard to their churning.
Maybe the author chose "churning away" to avoid the unflattering implications of "churning out." If so, I think it was a bad decision, since "churning away" does an exceedingly poor job of conveying what (I suspect) the author means. I would have suggested ending the sentence as follows:
...where the women of the home give voice to dreams and fears, and where elaborate social and political intrigues play out."
An author determined to tie the sentence to a kitchen-friendly verb could have used "cook up" or "brew up." Both have weaknesses of their own—"cook up" suggests fabrication of a deceptive kind, and "brew up" has witchy connotations—but at least they aren't borderline nonsensical, as "churn away" is.
Yet makes an exception to the hopelessness: despite his grim fate, the narrator takes comfort in his lack of panic.
For introduces an explanation of why this would be comforting.
But contrasts what he did not do with what he did. A similar usage of but: “The Patagonian mara is not an ungulate but a rodent.”
These three conjunctions do not bear any special syntactic relation to each other; they're not like neither … nor, for example.
Best Answer
"A place I could belong" is awkward, and you should probably avoid it. To see why, consider the following rephrasings:
You can't belong a place. You can belong at or to or in a place. So the most correct form would be "a place at which I belong" or "a place to which I belong" or "a place in which I belong", and since where implies at, "a place where I belong" is proper.
That said, "a place I belong" is used here and there. So are the other forms ("place we belong", "place they belong", etc.). If I look at Ngrams and squint, it's used as much as about a quarter of the time. If I search, I find "a place I belong" predominantly in song lyrics, though, and songs are known for taking a few shortcuts with the language to make the meter fit.