Ill help with what i can.
The heavybag, your aim is to focus on proper form, whilst focusing on throwing punches hard and FAST. I hit it when its coming towards me, this is my preference. When its going away then hyper-extension can occur so its not advisable.
Maybe fill your bag up with something heavier. I personally like the heaviest ones, but they do compact at the bottom and can cause pain when you forget this

With the rope, well, its hard to explain. Swing it, jump over it very quickly keeping your feet low to the ground, not like kindergarten skipping.
Once you get adept at both feet together then start alternating feet, hopping, jogging on the spot.....

You said you were a runner before so you probably know more about shin splints than me, but one thing i do know. Get them checked out. They can get worst and lead on to serious things. Maybe go to a proffesional running shop and get new shoes that are measured to your gait etc.