Thoma
This is probably too late for you but may help others. My daughters 5c showed exactly the same symptoms i.e. first wouldn't connect to network and I was told by network operator that i needed to update the iOS however when I tried to do this it froze when I got to firmware update and showed error code 50. Followed all the recommended procedures, phoned up Apple Support who simply ran me through the same procedures, took it to an Apple authorised dealer who simply plugged it into their computer with the same result!
About to give up on the phone when I thought would trawl the internet just in case. On one uninformative YouTube link someone suggested a hairdryer only to be derided by other posters. But other people suggested that it was a problem with the Baseband IC chip whatever that was. Into very technical videos taking your phone apart, some soldering on a new component others just heating the component, which brought me back to the hairdryer guy. So this is what I did:
Connect iPhone to computer and hold down Sleep and Home button for a full 8 seconds and then release the sleep button.
This will bring up the restore button on your computer. Now comes the bizarre bit, get a hairdryer lay your phone on its screen and heat the area around the sim card (opposite side to the volume buttons) and hit restore. Keep heating the phone whilst restore takes place. Once the phone has got past the firmware update you can turn off the hairdryer and let it complete the restore on its own. This clean restore then sets up the phone for new and I had network connectivity.
Sounds crazy but it does work. Once restore is complete the phone will tell you it needs to cool down before being able to use it. The bad news is I then restored my daughters apps from iCloud for the phone only to lose network connection again. So clearly one of the apps she has is causing an issue with the connections and we will have to go through one by one to find out which one. But this will at least get your phone functional again.
Hope this helps others