Thanks for reaching out to us. I will do my best to help with each part listed in your request.
Starting with the hinge, you asked if you needed the 10" or 14", however you said that the plate on the sill is 15-1/4". Are you referring to the hinge track? If so, we unfortunately don't have anything in that size. The closest option we have is going to be our 36-004-14, which has a 14" long track.
I would triple check that measurement just to be sure. I don't think 1-1/4" difference will be the end of the world as long as you replace the top and bottom tracks at the same time.
For the operator, you have either our 39-253LB or 39-438B, depending on the design of the guide at the end of the arm. The video below will show you what I mean. All you have to do is check under the sash or just detach the arm from the sash track to confirm.
We do sell the handle/cover assembly you show here, which is our
39-375. If yours is in good condition, however, then you can keep using it without issue.
Finally, you mentioned missing a 45 degree angled arm, but I'm not sure what you're referring to. I don't see anything that's missing from your window. The parts you have here just look to be corroded. Can you elaborate?