If I start a chapter or finish a chapter do any affinity missions, regular missions, segment updates disappear? Or is all content available all the time (after it is unlocked?)
Yes, there is a time limit on one particular character:
Lao leaves after chapter 9, so if you want to do any of his affinity missions, you will want to do it before then. His affinity is easier to raise than other characters, thankfully, so it's less of a grind with him.
The rest of the content appears to be perfectly doable at any time after it's made available.