Not a very difficult hike and it's easy to find walking trails where needed.
It's pretty easy and very interesting to walk along the Great Wall from Laolongtou to Jiaoshan. On the way you can stop at the museum and the first pass under heaven. The museum ranks among the best of all Great Wall museums.
Laolongtou to Jiashan must be the easiest of all Great Wall hikes and it's very easy to figure out where to go at the few places where it is not possible to follow the wall. If the air is clear (maybe you are lucky) just walk towards the mountains.
It's a good idea to get down from the wall here and there and see it from its sides rather than just from the top. And there are a few places where you must descend and make your way alongside until you can rejoin.
Photos from a hike along the Great Wall from Old Dragon's Head to Mount Jiaoshan
First views of Jiaoshan in the distance
Layers of bricks visible on main wall
Jiaoshan clearly visible ahead
Third bridge probably better to go around
Photos are from September 2010.
I would wait until spring before heading to Shanhaiguan. I went there in the wintertime, and although all the walls are still open to tourists, everything else shuts for the winter... including public bathrooms, restaurants, shops, etc.
I would continue up Jiaoshan rather than just to its start. You'll have plenty of time.
If you want to hike more in this area go to Jiaoshan, Sandaoguan (beautiful but not a very good place for hiking), or Jiumenkou.
It's easy to walk from Laolongtou to Jiaoshan, but beyond there to Sandaoguan it is quite dangerous. But it and Jiumenkou are easy trips from Shanhaiguan. You could commute further but it would be easier to stay local as you progress to the west. Small guest houses should not be hard to find along the way. I don't think you can make it from Jiumenkou to Dongjiakou in one day. That's a long and demanding trek. Dongjiakou to Banchangyu would be nice.
Shanhaiguan time lapse video – Laolongtou to Jiaoshan