Welcome to PW !!
First of all need to fix the itinerary a bit.
8th Aug, Thu - Day 1: Departure from Lahore - Please leave 7th night in order to reach Chilas by end of 8th August
9th Aug, Fri - Day 2: Arrival at Karimabad (Can be 30th Ramdan or Eid) - You will have some time so u can go to the forts etc.
10th Aug, Sat - Day 3: Karimabad to Khunjrab and back to Karimabad (Can be Eid 1st day or 2nd day) - This cannot be done in one day so u will have to stay at Sost or Passu on your return.
11th Aug, Sun - Day 4: Ultar Trek - It is rocky terrain and water availablity is almost none except gray mineral rich water of Ultar glaciers. Instead i would suggest to come back from Sost / Passu and go to Nagar Valley and visit Hopar Glacier.
12th Aug, Mon - Day 5: Karimabad to Naltar and Naltar to Gilgit - Reach Natlar as early as possible and trek to the bigger lake on left side where very few people go. This can be a reasonable trek for a family plus good explore too !!
13th Aug, Tue - Day 6: Departure from Gilgit
14th Aug, Wed - Day 7: Arrival at Lahore
Sost has hotels and better place to stay as compared to Pasu. Not sure if Gulmit has any hotel.
Shimshal - If your days are fixed then u can either go to Shimshal or Khunjrab. Shimshal is jeep drive of 8 hours return journey and if u do sight seeing and stay in Shimshal for some time then it can easily take 10-12 hours.
So, choose between Shimshaal and Khunjrab though i would suggest to do Khunjrab
Your travelling days are actually pre tourist season as on Eid days, these places will be almost empty and rush will build up on 3rd or 4th day of Eid.
Transport is always available as lots of people are travelling to their native towns for Eid holidays.
Be well prepared for Khunjrab visit as it is very high altitude and will be very cold. Keep an eye on your kid's breathing performance while you are on your way to Khunjrab from Pasu / Sost onwards. Normally kids above age of 6-8 months perform just fine at high altitudes but still be conscious.
Rest of the locations are just normal stuff.
Best of Luck for the trip !!!