The shaft from the motor that the wiper arm connects to sounds as if it a seized which is commonon on rear wipers because of the crap thrown up and lack or use. The motor will try and self park every time you use the car which will heat up the motor eventually burning it out. You need not just to switch it off but remove the wiring to the motor to prevent damage untilyou get the time tosort it out. Then when you can get the shaft sorted, you could try WD40 on it but it will probably require the gearing to be removed and all greased up. You can get a cover that prevents water from getting to the shaft however there should already be a sealing cap beneath the nut that clamps the wiper arm onto the drive shafter stripping down. Wd40 may shift it and allow it to work but this will soon dry out and eventualluy the shaft will stop again, spray grease will be better and could prevent further If it does then fine but let the motor run to ease out any ruston the shaft and tube from the motor. Don't leave it as you could end up burning the motor out and even cause a fire.

I think the light is a coincidence and is probably caused by corrosion in the bulb connector. Was the bulb discoloured eg blackening, if so then water is geting in and shorting it out. Again becase the lights arelow dwn they get blasted with water and dirt from behind and can be a problem to remove. WD40 sprayed down the bulb connector anda cleanup of any rust shold cure it. Are you geting potive at the cente terminal of the bulb holder? Yes then check for an earth (negative) on the casing perhaps the bodyof the lighting untis ot providng a stable earth because of corrosion. If you aren't gettingthe positive feed then check the multi way conector wich is behind the light unit on the loom crossing the rear of the car. Again te position of these puts them in direct line of water ad dirt spray and can cause corrosion, WD40 on these will usuallycure these. To separate the joint a clip is located on the side, the clip is pressed in whilst the two parts are pulled apart.They can be stubborn but take your time and they will separate. Check for supply on these and WD40 bot sides before rejong. You ca cover these in water proof grease (PH GREASE).

There is also a possibility that if you have a towbar fitted that SCOTCH CLIPS have been used to splice the tailer connector wires into the loom. These things are very good at corroding the wires and cause failure both of the trailer and the main car system. The offending clip sould be removed and a more appropriate connection substituted. Basically keep working back till you find the power break, however I think you will find it in one of these outlined areas,hopefully the first.

Both problems are typical in wet winters where dampness gets into insulation and corrodes.

I hope this helps get you started

OLDMINIMAN