On the oil question...once a year or so, depending on how much you use them...if you use them a lot, six months or so...
With the "skinny part/shoulder", it looks like a Singer shuttle...that doesn't mean it won't fit your S & R...comapare it to what's in the machine.
Your treadle likely takes a square or triangle key...try a tiny screwdriver to see if that works...if it does, take a flashlight and look inside to see if the works are triangular or square...you can find the keys online... :)