June is best to go on these places. Regarding the roads, the roads are good but in some area there are patches which are very bad. Regarding your car it is a new model civic 2013, i don't want to create a hesitation but let me be clear that civic has a low road clearance and you have your family with luggage with you this will cause some difficulties in some places only. Otherwise if you have more days then you should drive slow and reach there. You should at least plan for 10-14 days if you want to visit the places you have mentioned. More days are required because travelling time is quiet a lot. Regarding the places to stay please note as under;
In Besham - PTDC is the best option, rather its wonderful to stay over there
After Besham good hotels are in Gilgit, Hunza and Skardu
For staying in these regions please visit following pages to have an overview;
Serena Hotels | Gilgit-Baltistan Collection
Reviews of Hotels, Flights and Vacation Rentals - TripAdvisor (In this website mention the destination and view people comments on it, this would guide you alot)
www.eaglenesthotel.com