Disconnect your battery before doing this wear vinyl gloves, use a torch, get your glasses to see what you're doing 🙂 Get the panel off the inside of the boot door, take a look at the cables, anything is obviously is out of kilter, consider if this is the issue. Also note any corrosion as this might alter the grounding/earthing of the wiper motor. Where does it earth? If the cable whizzes off somewhere, see if you can work out where and check that out.
As far as the half way thing, it could be that the worm gear (if it's that type) in the wiper motor might be worn out, or the gear that it turns. Have you felt it? Does it really push all the way like it should or is it definitely electrical? Wiper motors are ten a penny, and easy to replace. As for the starting issue, not sure. Can't surmise without more info. Take a look under the battery, I noticed only today that in that cavity water can collect and cause problem perhaps.