New answers tagged noise
From your question, the whine is linked into braking both when the veicle is stopped or moving. I would first look at the brake servo, its hoses and valves. A vacuum leak can give a whining like noise.
Yes, the normal route is to add sound deadening material inside the doors and/or on the floor of the car. A corolla won't have much stock insulation so you should be able to make some good progress in reducing it. This is a common project for people who are installing nice audio systems, so you should be able to find some nice walk-throughs on audio sites. ...
Look around the battery terminal and the fender and see if there are any sign of arcing. The loose battery may have shorted to the frame killing the ignition power and then moving away restoring things to normal. The black smoke may be from the unburned fuel drawn thru the carb causing a rich mixture on restart.
Where did you buy the rotors? They could have been sitting for a while. Sometimes when you buy rotors from a discount auto parts store it is a really good idea to have them shaved a little first. Especially the cheaper brands.
Without any other symptoms or issues, I would assume you just ran over something that your tire threw into the body of your car. Like a piece of a re-tread tire. Look under the hood for anything that looks out of place and check under the car for any fluid that might be leaking. If you don't find anything just keep driving and pay attention for anything ...
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