Neelum valley is the most beautiful place I have ever visited. 4 days are not enough to explore complete beauty of this valley. However, see below answers to your questions.
1. Road condition is fine. From Islamabad you should travel via Muree Expressway till Lower Topa, from there on the road is single but pretty OK with few bad patches uptil Kohala. From Kohala to Muzaffarabad, road is normal as traveling straight, road is under construction at some places, can experience few bad patches. From Muzaffarabad, you will travel on Neelum road with the river on your left side until Nosehri from where you cross the river and then onwards river remains on your right side. The road condition is fairly good and completely carpeted uptil Keran (Lower Neelum) which is a little ahead of Authmaqam. Uptill here you can travel in any vehicle without any problem. From there onwards, road is not fully carpeted and there are certain bad patches which can worsen in rainy season. So, four wheeled vehicle is recommended or you can hire a jeep from Keran. Keran to Shardah is somewhat better than shardh to Kel.
2. Coaster and Hi-ace can easily go till Keran while Hi-ace can also go uptil Shardh if an experienced driver is driving it. However, it is recommended to hire a jeep from Keran to move onwards to Shardh and Kel. Walking is not much required. You can move along the river in the valley, until you want to climb on nearby mountains, then you might need hiking.
3. Route is like Islamabad - Lower Topa - Kohala - Muzaffarabad - Nosehri - Authmuqam - Keran - Lawat - Shardah - Kel - Helmet - Taobat
4. You should consider staying at Keran (Lower Neelum). AJK Tourism Department has a very well maintained rest house in Keran which is right on the bank of river Neelum while river Neelum is the Line of Control (LOC) at this point, hence the area on other side of the river is Indian Occupied Kashmir. Highly recommended to stay at this place. The next stop can be Shardah or Kel. Shardah is better to stay as the rest house in Shardah is very good.
5. You should get your tank full from Muzaffarabad. Fuel will be available at few places as you move on, but better to have a full tank when you leave from Muzaffarabad. Food is available at all places. Electricity is available in all places bot load shedding often occurs. Uptil Keran, facilities are better while good in Shardah as well.
6. Four days suggested plan is as follows.
Leave at midnight from Lahore and reach Islamabad early morning. Have breakfast in Islamabad
Leave for Muzaffarabad via Muree Expressway - Kohala and reach Muzaffarabad at noon. Have quick lunch in Muzaffarabad and leave for Neelum Valley.
Travel straight to Keran (which is around 2.5 - 3 hours drive from Muzaffarabad). On the way, stay for few minutes at Chalhana Post which is a passing bridge between Azad Kashmir and Indian Occupied Kashmir. You reach in Keran by evening of Day 1.
Take an evening tea at rest house after checking in. Then move upwards to upper Neelum (By foot or by Car), around 4 km passage, and you reach on top of hills. Spare your evening hours there. Excellent scenery. Move back to Keran Rest House in the evening for night stay. Book a jeep for next two days.
Day - 2
Leave in the morning in Jeep(s) for Shardah. It takes around 2 and half hour and you reach Shardah. Check in at Shardah Rest House. have your lunch and go out to visit beautiful sites in Shardah. Go in Old University, Visit some thousands of years old heritage with excellent breathtaking scenery. Return back in the evening. Have a night stay at Shardah Rest House.
Day - 3
Leave in the morning from Shardah to Kel. You reach to Kel in around 1.5 - 2 hours. Dont stop at Kel, Continue travelling to Helmet and Taobat which will take another 2-3 hours. You will reach there by noon. Then spare few hours visiting Taobat and Helemt and return back to Kel in the evening where you stay in Rest House at Kel.
Leave in the morning for backwards journey. From Kel, leaving at around 7 am in the morning, travel straight to Keran which you should reach by 12:00 pm. Have your lunch at Keran and leave the jeep there. Take your own vehicle and move back to Muzaffarabad. Leaving Keran at 1:00 pm, you should be in Muzaffarabad by 3:00 pm or 3:30 pm. Continue traveling back to Islamabad via Kohala and you should b in Islamabad by 8:00 pm from where you can have you dinner and leave for Lahore. Reach Lahore midnight.
I hope this itinerary will be much enjoyable for you.
Dont forget to share your experience later on.