From a muscles' point of view running is very catabolic for what I know, so maybe this affects also the arm/forearm muscles?
You certainly won't have lost it from the fingers in that sort of time. Your body is just adjusting. As you've just started up the running again, you'll have longer recovery times having more effect on your climbing. This recovery time will quickly drop off. That said, sleep well, eat well and make sure you take on what you need for recovery immediately after your run (for instance, a good glass of skimmed milk and a banana). In the long run, that level of running would do no harm, perhaps even some good (mainly if weight loss is helpful for you)
I think the only time running has ever affected my climbing is when I've been 80km+ per week or the day after a track session
I don't think it's inherently damaging for climbing, but from what I've read endurance training after strength training may reduce gains by ~20%, so they're not easy to manage concurrently, excepting the occasional burst of aerobic activity which is just healthy in my view.
I've been doing a bit of reading on shakes at MyProtein.com - they recommend more Carbs and rehydration for post running opposed to all Protein, which makes sense.