In an emergency, all the rules that used to apply, don't apply.
The idea that a flight attendant would ask someone to come forward and fly the airplane is a common fantasy that many people have, or have had. It's the quintisential hero fantasy, similar to saving the day in a restaurant when someone's choking, or stopping a bank robbery, or being the only person in the room who knows to cut the blue wire as the clock counts down to zero.
To address your questions, or points in order, bearing in mind that flight attendants don't yell:
1. Yes, nose-up, with no power change, the airplane slows down and gains altitude. Nose down, it can descend and increase speed. There's a lot more to it, of course, and there are airspeed and operating limits that mean one could stall or enter mach buffet by raising or lowering the nose at the wrong time. Doing so with the autopilot engaged may result in fighting the autopilot, which has actually lead to loss of control of the aircraft and in at least one case with a Chinese 747SP, a stall, spin, and permanent damage to the airframe. You've got the general principle, though.
2. Thin air makes for lesser engine performance, not better. The reason that we operate turbine aircraft at high altitudes has to do with efficiency; we get higher true airspeeds (how fast the airplane is really moving, vs. what the instruments tell us)...we go faster. The engines can be operated at higher power settings at high altitudes, and we can fly at higher true airspeeds without bumping up against aircraft limitations such as airspeed limits. Turbine engines need to operate at fast rotational speeds and higher power settings to be their most efficient (power output per gallon of fuel burned). At low altitudes, they put out so much thrust that the aircraft will go too fast, when operated at high power settings; thrust must be reduced, the engine operated at lower rotational speeds and power settings, and the engines are much less efficient. We generally try to climb high as quickly as possible, and stay there as long as possible, which allows us to go as fast as possible (within limits); this is to cover as much ground as possible for each gallon of fuel burned. It's a bit more complicated than that, of course, but you get the idea.
In an emergency, one usually won't be pressing on to the destination. Fuel economy may not be the priority. Then again, it might, depending on the circumstances.
3. With a depressurization at altitude, especially a rapid depressurization, it's important to get to a lower altitude as soon as possible. More important than that is that the flight crew get on their oxygen as quickly as possible, go to the emergency 100% setting immediately, and follow emergency descent procedures. Passengers aren't provided the same kind of masks or pressure oxygen system that the pilots have, and instead receive oxygen that's not under pressure. If the depressurization occurs at a high altitude (35,000', for example), chances are that the passengers are going to pass out anyway, on the way down, even though they have oxygen. They may have 100% oxygen, but there's insufficient air pressure to get it into their blood stream as they breathe.
Generally we try to get down to 14,000 as rapidly as possible, but it depends on the terrain below, as well as where the flight is at the time. At low altitude, with a long distance to cover to get to a landing site, fuel consumption could be a big issue with something like an oceanic crossing; the flight circumstances determine the proper course of action, and it changes with the needs of the flight. You're correct, however, that generally below 10,000, supplementary oxygen isn't needed.
4. Commercial passenger airline aircraft have much lower structural limitations than fighter aircraft so generally speaking, one shouldn't fly inverted or attempt vertical flight (or do loops or barrel rolls). Commercial aircraft are designed to fly fast, but there are limits (and they change with altitude and flight conditions); the speed limit as the aircraft climbs to high altitudes, or descends from a high altitude is actually a moving target; it's a mach operating limit that remains a fixed value, but a changing airspeed. It's still a limit that's much lower than some tactical aircraft, so no maneuvering to avoid phantom fighter jets while doing that depressurized descent to safety.
5. You've got it; reduce power for landing, though if you've flown commercially, you may have noticed that sometimes power is increased during the approach to land, or shortly before landing; power is used as needed up until reaching the runway. Then it's usually brought to idle, and the airplane landed. Power may appear to increase rapidly just after landing. In this case, the power levers have been brought to idle, and sometimes the same levers, or different ones, are brought into a "reverse" range, where the engines use thrust differently in order to help slow down. In this case, power actually increases again, but the thrust from the engines is diverted to the side, or sometimes even forward, to help slow the airplane. Cars don't have that feature. Brakes are also used, just like cars. In airliners, an automatic braking feature is used, giving a setting that slows the airplane at a specific rate. This heats up the brakes. When reverse thrust is applied, the aircraft still slows at the same autobraking rate, but less brake is used, so the brakes don't get as hot.
6. Pull back to go up, push forward to go down, and move the control wheel, or yoke, left and right to bank the airplane. None of those will turn the airplane, though. Pitching the nose up by pulling back increases the angle between the wing and the airstream, and that angle changes the lift created by the wing as well as the line of thrust, and that in turn changes the aerodynamics and creates a climb, assuming excess thrust exists for the new condition. The opposite applies for descending.
Banking, which is turning the wheel left or right, doesn't actually turn the airplane. Lift acts upward when the airplane is flying level, but when the airplane is banked, some of that lift isn't acting upward, but to the side to which the airplane is banked, and it "pushes" the airplane, resulting in a turn. The steeper the bank, the tighter the turn. Much of the time, airliners are restricted to about 25 degrees of bank by autopilot limitations, passenger comfort, etc. With small amounts of turn, less bank is often used.
Aircraft also have rudders, which don't turn the aircraft either, but do affect the turn to some degree. In small airplanes, the rudders are used by the pilot quite a bit. In large airplanes, not nearly so much, and their function is sometimes more automatic. They can be thought of as "quality" devices that improve the condition of the turn by accounting for something called "adverse yaw." Adverse yaw is caused by the ailerons and swept wings in a jet...the ailerons are moved by the control yoke when turned left or right. It's all interconnected; one thing affects another. A bit more complicated than a car, but you've got the general idea.
to be continued...