I roll my eyes a little bit at "core" exercises, because your body works as a full whole so if you're doing proper full body exercises your "core" (which isn't some agreed-upon anatomical term) will not only be trained, but will be trained proportionately. This is a huge deal and can't be overstated: a lot of training is down right dangerous and counter productive when you isolate areas and don't allow your body to function in an anatomically normal way.
This isn't to say that there aren't a variety of important often-forgotten muscles in the middle of your body, but that's more reflective of bad fitness regimes (think: guys who bench and bicep curl all day, or cardio-princes(ess) on the elliptical reading magazines).
Plenty of tried and true strength training practices offer incredible "core" development in safe, balanced, and effective way. Plus, they don't work in isolation so you get a pile of overall strength gains.
How long should a core stability session be? Is it better to do a
wider range of exercises or do multiple series of fewer exercises?
Even if your goal was purely strength training, you could get by quite well with three sessions of an hour a piece broken up throughout the week. Since you have such a consistent running schedule, I'd shoot for 2-3 times a week of 30-45 minutes.
You should always focus on a lower number of effective exercises versus a higher number for the sake of variety. You only have so much time to train: it's foolish to do anything other than activities that are the most effective. Here's an answer that addresses some particular exercises and schedules.
I'm also going to take up stretching sessions. How to join core and
stretching with my standard cycling or running? Should I do a few
sessions in a single day or reserve special days for core/stretching?
I would really look into yoga, iyengar if you can find it, but anything with a good instructor will be good. It sounds like you've got a pretty loaded up training schedule, so for yoga if you can't do it on your complete off-day, consider doing it as a warmup before a run, or right on the heels of a run since you'll be pretty limber at that point.