As Anil has stated, there definitely is a route that goes off NH8 from Kishangarh to Hanumangarh, Abohar, Amritsar, Gurdaspur and joins NH1 at Pathankot. Taking this route, you may also add Golden Temple and Wagah Border in your itinerary. Check the map here: http://goo.gl/maps/Nn6fZ
Anil & Aashish have provided you most of the answers. As for Day 4. Going to Pahalgam from Jammu via Doda-Kishtwar means a drive time of 12 hours over mountains. Not advisable – better take Ramban-Banihal-Anantnag route.
As you’ve stated that you have additional days in hand – you may want to add Yushmarg (If you’ve heard of Khajjiar, this is bigger!), Doodhpathri (no accommodation, do a day trip from Srinagar or Yushmarg), Wular lake, Aru alongside lidder river and Verinag too in your itinerary.
Oh Yes! Kargil won’t be possible as Zozila will be under several feet of snow, Sonamarg you can do and when you go there, do visit the Thajiwas glacier.