I've sorted out how to deal with the half stitch at the beginning of the cast on.
The fix is quite unbelievably simple, and I can't believe that I didn't think of it before. Anyway, better late than never - here it is.
This mess is the provisional cast on just before I start working in the round.
This is Opal Tiger yarn, as you might guess, 16 stitches on each of 4 needles. The yarn tail is the dark brown bit, and the working yarn is the orange bit. That sloppy thing at the end of the last needle is the last yarnover, not to be mislaid.
All you need to do - before you work the first stitch, make sure that the working yarn is picked up from underneath the yarn tail. Or, in other words, make sure that the yarn tail goes over the working yarn.
Literally - put the (dark brown) yarn tail across the (orange) working yarn -
- and bring it back down again, out of the way.
Then just carry on.
So, after that, I worked my picot hem, unzipped the crochet chain completely, and picked up the stitches from the lifeline all the way round - and there was exactly the right number, with no half stitches to keep track of.
Here you can see the first needle where the half stitch used to be, at the very beginning - with 16 solid stitches on it. Very gratifying. That dark brown end of yarn is still the yarn tail, by the way.
I promise that all the other needles also have 16 solid stitches on them.
Here is the wrong side of the work after I've finished joining the hem, just in case you are interested. I do like this method, in fact I can't see that I would ever use any other method for hems now.
And now I think I shall go and knit some more sock, actually.