Like many native French speakers, I pronounce the vowels of ceux, sœur and ce differently. /ø/ is more closed than /œ/ which is more closed than /ə/.
My prononciation mostly follows the phonetic classification, but not completely. There is some context-based or free variation. The “schwa” sound /ə/ tends to drift to a realization of [œ] or even [ø] when it's emphasized. /ə/ can drift to [œ] when followed by another [œ] (“le cœur”: [lə.kœʁ] or [lœ.kœʁ]). On the other hand, “le deux” remains [lə.dø], I wouldn't pronounce the two vowels identically. /œ/ can drift to a realization of [ø] when followed by another [ø], for example heureux is in theory /œ.ʁø/ but I (and I think most French speakers) often pronounce it [ø.ʁø]. On the other hand, in “heure creuse” [œʁ.kʁøz], I don't think I'd ever pronounce the two vowels identically: the consonants in between are too complex for the proximity of the vowels to matter. And likewise, “deux cœurs” remains [dø.kœʁ]: the influence doesn't work in this direction. I think /ø/ is stable: it's always realized as [ø].
Not all French speakers pronounce in the same ways, as shown by examples in this thread.
The standard analysis of French phonology is based on a kind of “middle ground” pronunciation which is roughly an early 20th century urban pronunciation from the Loire valley (i.e. south of the North of France). This analysis results in 16 vowels (/ɑ/, /a/, /ø/, /œ/, /ə/, /e/, /ɛ/, /i/, /o/, /ɔ/, /u/, /y/, /ɑ̃/, /ɔ̃/, /ɛ̃/, /œ̃/). Most French speakers don't distinguish those 16 vowels (I think 13 or 14 is common), but which ones don't exist as distinct phonemes depends on the speaker, mostly based on regional variations. For example, /ɛ̃/ and /œ̃/ (brin/brun) are mostly indistinguishable in the north of France but distinct sounds in the south. On the other hand, in most of the south of France, /ə/ and /ø/ are not distinct phonemes. Conversely, there are vowels that don't fit in this analysis, such as /ɛː/, which is typically a distinct phoneme from /ɛ/ (fête/faites) in Québec French but typically not France.