Being high is actually safer than being low with no engine. The DA40's airframe is designed from a glider and at 12,000 feet with no engine, you can glide tens of miles or more, enough to get to an air strip in most cases. Plus it gives you more time to react and do everything on the emergency checklist correctly. In fact, compared to the Cirrus SR22 (a.k.a the "doctor killer") that has a parachute, the DA40 in a full stall will loose altitude at a rate of 300 ft/min, whereas the SR22 will fall at 500 ft/min with the parachute deployed.
Being low and slow will kill, such as if an engine quits right after take off. If take-off cannot be abandoned, then you're only option is to land straight ahead regardless of whats there. If you turn around, you'll loose air speed and altitude and may stall and crash to the ground. In this case, its a no-win scenario.
As for general redundancy, there are a couple of things that come to my mind. These apply to most popular general aviation aircraft:
Most use engines that are 4-cylinder horizontally opposed. For each cylinder, you have two spark plugs operating independently of each other, for redundancy. This is reflected on the ignition switch; you've got "off", "Left", "Right", "BOTH", and "start". "Left" and "right" refer to the two spark plug systems. Before take-off, you make sure that the engine can operate correctly at a certain RPM with both or only one spark plug systems. Typically you have the ignition set on "BOTH" but in the case one fails or is choking up the engine, you can isolate the problem by switching it to "left" or "right" only at the expense of slightly lower engine RPM.
You've got redundant batteries. Two for the engine (24v system), one for the emergency locating beacon (ELT), and more for other things. In the DA40, you've got four batteries: two for the engine, one for the ELT, and finally one emergency pack for the avionics (that big, red covered switch near the dashboard) should everything else fail.
On top of the batteries and the main alternator, you've got two "magnetos", one per spark plug system. These are essentially mini versions of the alternator; once the engine starts, the magnetos generate power to the spark plugs. At this point, the engine can run smoothly without the engine batteries. That means once the engine starts, I can shut off the main power switch and totally forget about all the batteries (but this is not recommended, for obvious reasons). The reason for this is to have isolation between the engine's electricity and everything-else's electricity. Note that if you do that to a car, your car engine will die because its dependent on the battery.
Unlike a car, you've got circuit breakers instead of fuses. I can turn off the aircraft's main electricity to certain parts of the avionics to isolate a problem should one occur.
In the DA40 and all aircraft with digital avionics, you've got backup analog steam gauges should the avionics fail. In the case of no electricity, that means I've already tried restarting the alternator, checking the alternator charge amps, turning off un-essential equipment, checking the circuit breakers, switching to the essential electricity bus, AND trying that big fat red emergency avionics battery backup.
You've got redundant static and pitot ports, which are basically sensors to measure air pressure for the air speed indicator and altimeter. In the DA40, you've got two ports should one get obstructed by debris, dust, or bugs. You also have a heater in the pitot so it doesn't turn into a chunk of ice when flying in clouds. On a larger plane like Boeing, you've got several of them all scattered throughout the body of the plane.
In any aircraft whose primary avionics is digital, chances are it will have two displays, like the DA40 here. Each screen is actually one independent system, but since there is two, both of them are connected and work together. Should something happen or should one screen die, I can hit a switch and the two screens will operate independently and display the same information.
You've got two radios, two GPS receivers, and two VOR navigation receivers.
But the back-bone to all this are the maintenance crew. Unlike servicing cars, one does not decide to just open up a "plane service shop". To legally service any aircraft, you need a license called an A&P license from the FAA, which is a career on its own. Pilots can only do preventative maintenance, such as filling air into a deflated tire or add motor oil into the engine.
There are also strict regulations towards aircraft inspection. Log books are kept for the airframe, propeller, and the engine. It is completely regular for mechanics to refuse service to airplane owners that do not have the log books or have forged log books for various reasons. The engine needs to be overhauled after a certain amount of hours, which is basically taking it apart, cleaning and inspecting every little bit, and putting it back together as if it was new. There are also requirements to have the airframe and transponder inspected annually or after a certain amount of hours.
Now, having said all this, I still want to emphasize that flying is a managed risk, like driving. Is it the safest, even with everything listed above? No. But, like driving, certain precautions are taken to reduce the risk.
In regards to multi-engine planes, they are more expensive and you do need another level of license to legally operate one, but there are those that argue on the grounds of safety. Personally I see no point in taking a twin-engine general aviation aircraft for a domestic flight above ground. There are those who fly across the ocean to the Bahamas, in that case it would be logical to prefer a twin over a single.
I hope this will at least clear some things up and hopefully alleviate some of your fears. If you're into aviation, go to your local airport and find an instructor or flying club, sometimes they give discovery flights for free.