From your description it is definitely the starter motor sticking in the engaged position! Probably needs replacing, either the release spring is broken or needs lubricating!
wow , never turn you wipers on when the cars has snow or ice on the window EVER , you have to loosen the nut that holds the wiper on and re-adjust it to the other one , easy fix unless you work out the knurle that is what holds it in place easy fix ,unless you broke it
There normally mounted really closely to the fuel tank. Look for hoses leaving the tank and going to the canister normally towards the rear of the vehicle.
the battery may appear to be ok but if starts with a jump then the battery probably has'nt enough amps , get a new battery and check the charge rate from alternator
well you could cut the dinger out.. but I would not recommend doing that. Usually it makes more sense to just wear your seat belt as local laws require
Not usually, when you have two O2 sensors in an exhaust one will be heated and the other not. Ring Hyundai and ask for a price for each sensor, then get them to give you the part number for reference.