LMAO at so much of the crappy 'advice' given in this thread! It really is shocking to read some of the replys!
A lot of your questions are subject to the age and type of car you are driving. Even the engine type (V configuration, flat, boxer) will effect some of the appropriate answers. Being an FIA licence holder I will base my answers on a modern performance car setup and the type of driving you would expect from an average proficient driver.
1. Can one take the transmission out of gear (from gear x to neutral) without disengaging the clutch (ie, stepping down on the clutch pedal)? What does it do to the transmission.
Yes, you can take a car out of gear without using the clutch, but there is no real reason why you would want to do this. Assuming your car has a single clutch setup with syncromesh on all gears, if you did need to drop the box into nutral without using the clutch only do this at low RPMs.
2. For making a turn (where one has to slow down first), how does one manage the gearing? Coast in neutral into the turn, then engage appropriate gear once the smallest radius of the turn has been passed?
You should always have your cornering speed sorted before you enter the corner. Keep your car stable by braking with your front wheels as strait as possible and drop to an appropriate gear before you enter the corner. Balance the power of the car through the bend and 'drive' the car out of the bend by accelerating through the exit. You should never coast while braking.
3. I always use the brakes to slow down, and rarely downshift. Best practice for slowing down and stopping?
Yes the brakes are the most important component in slowing a car down, modern cars don't require you to engine brake as much as they used to. However, it is important to change down the gears as appropriate to get the best control out of your car. For example if I was slowing down to stop at a set of lights I will change down as the car decelerates matching the engine speed with the ground speed.
4. To optimize fuel efficiency, I always want to be in the highest gear possible in order to be at the lowest RPM?
Not true. This depends on the stresses imposed on the engine. The engine is most efficient at peak power. Some engines, mainly diesels peak at a lower rev range, say 2500 rpm, whereas a high performance V12 may peak at around 6000 rpm. My car offers max torque at around 4,500 rpm. If you are accelerating it is important to change to a gear that allows the engine to work efficiently. Cursing along at 50mph in 6th gear and then flooring the gas pedal to overtake someone will send a surge of fuel to the engine that will be wasted before the engine builds up speed and has enough torque to let you pass them. It is much better and more efficient to drop a gear or two before you put your foot on the accelerator to pull away.
If you are driving quickly it is good practice to 'blip' the throttle as you disengage the clutch to match the engine rpm to the transmission speed before selecting your lower gear. With practice this can be done in one smooth motion and even done under braking (heel-toe).
If you are cursing along the motorway with little strain on the engine, then yes you do want to be in a high gear at low RMP. Make sure you do not let the engine struggle unnecessarily though.
5. 2 hands or 1 hand on the wheel? Or 1 hand on the shifter at all times?
This again depends on where and how you are driving. It is good practice to have 2 hands on the steering wheel as much as possible. There is no need to have your hand on the gear stick at any other time than when changing gear. If you are on an empty motorway crusing along at slow speeds then there is no reason why you can't relax a little!
Best hand position on the wheel is to place you thumbs over the top spokes of the steering wheel and wrap your fingers around the back between the 'ten to two' and 'quarter to three' position.
6. Near redline shifting, who's good enough to it, and has there ever been an occasion where you did it.
As mentioned before, different engine types provide best performance at different stages in the rev range. If you just want to pootle about economically, then there is no need to take your engine up to red line before changing gear, just change where you feel comfortable in your acceleration. If you want to drive quickly, then use your full rev range! The variable valve timing in Honda V-Tech engines doesn't kick in until after 5000 rpm, so you have the most fun between 5000 and 7000 rpm. My 997 provides a similar experience and when on the track I always take the car right up to around 7900 rpm before changing up. You shouldn't really red line in first gear, you will usually just bounce off your limiter and it will slow your acceleration. In fact you should try to never hit your limiter at all.
When changing down your gears to hit peak torque to overtake someone, or before entering a corner, make sure you chose your gears properly. This just comes with practice.