Two problems!

First problem sounds like you have a vacuum leak possibly at the brake booster.It retains vacuum after the car has shut off so it could be leaking through a rubber seal.Or this could be a fuel line leak even.Do you ever smell gas or hear any hissing noises? I can't think of what else it could be.Sounds like something pressurized leaking against a rubber seal?? Is it loud? Very high pitched?

Second problem sounds like pads.Is it a slight humming sound mixed with a a slight scraping noise? Check your pads and rotors on the back.Make sure you have some usable pad left,and not rubbing metal to metal.I might also add that it is normal for a 'little' bit of humm or rubbing noise to be heard,scuffing especially if the rotors sat for a couple days and got wet.It's the rust of rotor surface oxidation that built up and is causing scuffing noises to be heard at low speeds.

Make sure you clean off that rust and added a generous ammount of caliper grease to those pins.Those are what allow the caliper to move freely,mainly for the outside pad.But if one cant move too well,it stops up the caliper.

Do you live in the north? Perhaps that would explain the rust.Either that or the brakes were riding the rotors and baking the metal.Both will cause nasty corrosion.

About the squealing noise your hearing.The booster was more of a suggestion.Something to check on.No its not hard to get to.You know where you add your brake fluid at (master cylinder) , The booster is the big black round thing underneath that.The sound 'could' be coming from the hose that connects on to it.The booster is constantly under a vacuum from the engine running.The booster stores a small amount of vacuum.When you shut the car off,the engine stops sucking (no pun intended) and any leak that is present will start equalizing the pressure from air being able to enter.Just a thought.Since it is random,a hose or something moves or vibrates from the previous time you started/drove your car.temperature causes things to expand & contract.Again,do you live in a very cold area?Since you aren't smelling fuel,I would doubt it is a fuel leak.Since it only happens briefly and is intermittent,it could be something that is only under temporary pressure (ends when you turn off engine) and unlike a radiator hose which would still retain lots of built up pressure for at least a half hour afterwards.

I can only guess right now.I cant tell you what is causing the noise.Do you have antilock brakes? How many miles are on it? Try this,open your hood and start the car.Let it run for about 15 minutes.Then,with your drivers window down,reach in and shut the car off and very quickly walk around to the engine area and see if you can tell where the noise is coming from.You may have to restart the engine and and then shut it back down after 10 seconds.Thats how I locate intermittent noises like that.