good to know that you are thinking of neelum valley after seeing my pics 
here are the answers of your questions in the order you posted them:
its 10-12 hrs journey from lahore to muzafarabad
yes, you can go till Kel on your car, after that its jeepable track till taobut...
petrol will be available in towns like athmuqam, kundal shahi, sharda and kel... not sure about cng... its better to make your fuel tank and cng completely filled from muzafarabad...
road condition is pretty bad
muzafrabad to sharda (7-8 hrs).. sharda to kel (2-3 hrs with stays on picturistic spots)...kel to taobut (4-5 hrs with stays)
reasonable accommodation is available to stay even with family in athmuqam, kundal shahi, dowarian, sharda and kel...
weather is very pleasant.. little cold at night..
you can travel with family...
post below is high recommended for you
https://www.pakwheels.com/forums/road-trips-vacations/142596-exploring-neelum-valley-10-13-july