A good news update - a surprise iphone problem fix. My phone is working perfectly and I didn't have to send it away for 2-3 weeks to a phone repair center, which is what was suggested by the repair shop, because staff were on holidays (Scandinavian summer time.) I took my phone home to consider my options in posting the phone to a repair center and continued to fiddle with it, trying to get it to charge up. At one point, quite by accident, I held the power button on, WITHOUT the home button pressed and WITHOUT the power plugged in and suddenly the Apple Logo appeared. I simply continued to hold down the power button until the full functioning home screen lit up, displaying 100% battery charge. I have to admit I was surprised, after days of fiddling and frustration and a day spent driving to find my repair options.
Here's how I can best explain the fix. The various attempts to charge the phone (over 2-3 days) had finally, finally resulted in a full charge. It was only when the phone was fully charged that the reset would work (using Power button hold). Once the reset happened (Apple logo comes on) the iphone started to properly recognize a charger. It took me a day to trust that the phone was going to continue working and charging and it has now for 3 days.
One solution to the problem of battery hanging at 1% (or whatever) is doing a physical reset. However, an important missing piece of information is that you must somehow ensure that you drip feed enough charge into the phone so that the battery is 100% FULL (though you can only guess when this happens), and then attempt the reset. Perhaps Apple might add this key point to their repair suggestions.
I've lost my naive ideas about the reliability of an iphone, but have learned some new survival solutions. When working or visiting remote locations (i.e. no established wifi) I will once again to bring a back-up phone OR a tablet with a 3G option. Before travelling, perhaps I'll also remember to reset my phone and perform any needed software updates, while I'm still within striking distance of an Apple store or a repair center.
Thanks for the many clues to finding solutions from users on this thread.