Hello and Assalam o Alaikum,
I have done pretty much the same trip in the past with my family, with a mom who needed a trip due to high blood pressure and we managed to do a pretty decent one while every second was enjoyed.
We had a Corolla 2011 with 9500km on odo but with dunlop tires, not the local euro-star ones.
Firstly, if you are travelling with a family, you dont need to hurry. You need not to hurry at leave by 5.30AM. Sunrise now-a-days are at 6.30AM. If you are planning to leave with a family, you can easily leave at 7. People suggest to leave early but that is because of 2 reasons. They can reach Lahore in one day (21 hrs usually) with 3 stops but since you can stop at RYK (Faran Hotel is pretty good with proper parking with car wash 
etc, good rooms, AC and food) and second reason is that people suggest to travel in daytime in Sindh. Its alright to travel at night too but u cant make stops in case of emergencies. If you leave at 7AM in the morning, it would take about 11-12hrs to reach. Its convenient for your family to get ready by 7 as your wife need to prepare food aswell. Home food is advised although there are a few hotels at which u can stop. If you want to breakfast out, so that your wife is free of some work, u can go to Hyderabad and have a quite nice breakfast there, Subway on Hyderabad bypass is decent (for breakfast only). You can stop at Moro, to get fresh drinks i.e about 5.30 mins drive from Khi. New road at have been paved at Khairpur. Karachi-hyd is just ok till Nooriabad, then its better until Moro. You would be at Moro by 1.30PM. Then Sukkur by 4PM. and onto Ubaoro by 5.30 and Sadiqabad and RYK by 7PM.
After you leave RYK next morning, i would again suggest at 7 or 7.30AM after breakfast, it would take about 10 hrs onto Lahore. Punjab roads are better in shape but often very crowdy. Its totally safe to travel at night. You can stop at Ahmedpur East, Multan or Khanelwal and Sahiwal for snacks. Dont go via E5 Expressway which would say LAHORE on the board. Go to Multan and onto Lahore as E5 is in worst shape and very hectic althought its short 40kms but it takes same amount of time. Road is pathetic. Go to Multan and onto Khanelwal and onwards. Stay on N5.
Its safe to travel at night too, i personally like to drive at night as its easy to communicate with other drives on road via your car lights and its less crowdy.
On N5, you can drive upto 120kph, a steady 110kph is decent speed on a Corolla. I would suggest a tire pressure of 28/29 psi with car empty. Police is usually after the slopes with speed cameras. I guess, 800PKR is fine for over speeding in a low speed zone. It would say AGAY ABADI HAI on sign boads which limits at 70kph and they are usually in that zone.
You can stop at CNG pumps as they are cleaner than normal petrol pumps. You can get highway code book from any Post office and if you want more security from police, u can get ACLC NOC letter from ACLC office from i guess Gharibabad underpass side.
If you feel sleepy, dont try hard to take rest. Step out of the car, get tea, redbull or energy drinks to revive your self. If you get tired, dont try hard again and stop wherever convenient.
On n5, if a cow, goats or a dog usually comes in front of cars and people make emergency brakes and in order to save the dog, they crash badly. Cows and goats come rarely. If a dog comes in front of you, dont brake. Although it sounds pathetic and it would damage your car bumper, ac and may be even radiator but everyone's safety is priority.and Allah knows it. Trawlers do come too, you need to be active.
Onto ISB, you can go from Motorway (GT Road is a bit slow during the day time), it takes same amount of time but its 290kms from GT Road and 370km from M2. I take the motorway due to its proper road, less traffic, proper stop-overs. Speed limit is 120 but 130kph wont get you any tickets. I drive at 130kph steady and it would take 3.15-3.30 mins to reach with 2 stop overs. Although 130 seems alot but on Motorway it seems slow. Use 5th gear throughout and corolla would give u a 15km/l with ac on both N5 and M2.
If you leave at 7AM in the morning from Lahore, you would be at in Murree by 1PM at most. Check-out takes place at 12PM midday so, u can find hotels at cheaper rate during that time.
Seatbelts are must for every passenger. If you have a good babyseat or carry cot with a seatbelt holder, you can keep the baby in the carrycot and attach seatbelt to carry cot in your backseat. Carry-cots have seatbelt holder at sides, and u can put on seatbelt on the baby aswell which is in the carrycot itself. Google it. Tell your wife to get wet tissues, wet nappy cleaners and a lot of plastic bags. Keep the highway clean too.
You can get snacks such as sandwiches, biscuits, chips cold water in a cooler for travelling and can get ice too from vendors as water gets hot quickly in boot.
At times, due to congested N5, it gets hectic when trucks come in front of you. If u have a active passenger in front seat, you can ask there help always, for a over-take from left most lane (the emegency lane), ask if there is any truck or car standing or its clear, u can overtake if the pasenger is active. If they are out sleeping, dont try at all.
You can get a tow rope, Torching wire, complete tool kit with a little wire and a handy car tire pump which works with a car battery although in a new car its optional.
As for the Nathia Gali scene, you can travel safely via Murree Expressway, its new, 4 lane road, 2 on each side. keep in lane, keep in low gears, 3 at most. 4 gear can be applied on Murree Expressway though but not for nathia gali etc. Never change lanes on single lane roads or in blind corners. keep to your left throughout. Dont change lanes or take cuts as we do in karachi. 40-55kph is ideal speed on a Corolla which is controllable.
As for getting back to KHI, you can come use same hotel, stop-overs and staying on N5 throughout and M2.
Just a personal suggestion, you can go uptil Naran-Kaghan with ease on a Corolla. Dont hesitate to go there too. Would cost 10K more but would be more nice. Proper roads are there too.
