Swat and Kaghan Valley are different places though they can be included in one trip but connecting travel of 1-1.5 days is required from Naran in Kaghan Valley to Mingora in Swat Valley.
Your car can easily go till Jalkhad which is ahead of Naran. If u find it easy to drive in Galiyaat then there shouldn't be any issue in driving in Kaghan Valley.
You need to hire jeeps for Lalazar, Saif-ul-Muluk and Sri Paye.
Ansoo Lake is 12-14 hours of return trek at high altitude from Saif-ul-Muluk onwards and it would be too tough if u are travelling with family. Though u can visit Malika Parbat basecamp on horses.
Regarding security, there isn't any such issue till Naran and areas before that. Just avoid Lulusar Lake and Babusar Top if u are too concerned about security as recent incidents of looting and killings happened in areas beyond Lulusar Lake.