1) After mid june, definitely yes. However, if there is any land sliding on any particular day, it gets close. But in june its most likely because most land sliding happen in monsoon season (i.e. Late July/Aug).
2) There was some sectarian violence in the past in Babusar, Kohistan & Diamer. However, no such incident happened last year especially after launching of Karakoram Task Force which protects the convoy on KKH.
3) In G-B main cities like Skardu, Gilgit, Chilas, Astore etc, all cellular networks work. However, in far flung areas, the best coverage is by SCO (Special Communication). Its a cellular network operated by Army and if u want, u can purchase its SIM as soon as u enter GB.
4) Hotels are very reasonable in GB. No looting like in Galyat or Kaghan valley. Search for the best option on the spot. You can even camp at many locations like Rama, Deosai and its perfectly safe. Jeep rates will depend on your itinerary.
5) Its working in Gilgit. Not sure about other areas. However, hotels in Gilgit, Skardu and Astore offer Free Wi-Fi service. Ask the particular hotel before check-in
6) You can visit Dharamsar lake. Its near Babusar. And for G-B, you have many options like Rama Valley, Rama Lakes in Astore. Deosai in Astore & Skardu. Manthokha Waterfalls, Shigar near Skardu. Khaplu in Ghangche district. In Gilgit you can visit Naltar valley and Lakes. In Hunza you can visit Karimabad, Hopar Valley, Attabad lake etc.
Check out my thread, it will give you more info about the area and points to cover: https://www.pakwheels.com/forums/road-trips-vacations/221449-14-days-heavens-gilgit-baltistan-mega-tour-gb