Bro first of all thanks for the appreciation.
Yes you can go on honda city via KKH. (Some members visited deosai on corolla, for this you must of big heart :).)
Road conditions are not good but one can manage if drive with patience.
Lahore to Besham took 10-11 hrs roughly.
You can go with family. No issues.
Regarding security issue, while travelling I do not felt anything like this. So the advise is
"Just faith in Allah and keep rolling".
Aytal-kursi ka vird rakhin inshAllah khair hoo gee.
Wishing you good luck for your upcoming trip :).