What should I do to increase the lifetime of my crampons, maintain them properly, and check whether they can still be used safely?
There's one good way to make sure that your crampons last forever and are completely safe against failure in the field, which is to leave them in a closet and never use them.
Assuming that you are going to use them, you can try to avoid walking on bare rocks in them. Walking on rocks will dull the points and will also stress the metal by repeatedly bending and flexing it. Falling on your crampons can be particularly damaging. Black diamond says that when they get crampons returned with broken points, it's normally the front points. Of course, sometimes in mixed conditions it's impractical to take your crampons off and put them back on repeatedly, and this will entail a certain amount of wear on them. For mixed conditions, consider bring Microspikes either in addition to or as an alternative to crampons. Don't endanger your safety by failing to put on your crampons just because it's a pain to keep putting them on and taking them off; if you need them in a certain spot, you need them.
Crampons need to be sharpened periodically. To do this, you need to buy what's known as a mil bastard file. (The term refers to the roughness of the file.) When you buy a file in a hardware store, it has a bare spike, which is meant to be inserted into a handle. If you don't have a handle, you can hold the file by the spike. Hold the handle or spike in your right hand, and put your left hand on the far end of the file. Point the edge of the crampon's claw toward yourself, and stroke away from yourself, so that you're stroking away from the edge. Don't go back and forth with the file; only stroke away from yourself. If you have a card (i.e., a brush), you can use it periodically to clean the tiny bits of steel out of the teeth of the file. I haven't figured out a good way to clamp the crampons while working on them, although it sort of works if I attach them to my boots and then try to keep the boot still.
While sharpening your crampons, visually inspect them for cracks.
Rust is not a safety concern in and of itself, but it will accelerate wear on the teeth, which will force you to sharpen them more often and shorten the lifetime of the crampons. Sharpening will automatically get rid of some rust. If possible, dry your crampons after use, but of course in real-world conditions this may be impossible. If you have to walk along roads that have been salted, try to get the salt off as soon as possible. Some crampons are made of stainless steel, which is less susceptible to rust than the standard chromoly steel.
If there's still a lot of rust left after you sharpen the points, you may choose to try to remove it, but this isn't critical. To do this, try working on the steel with oil and a mildly abrasive tool. The abrasive could be fine-grit sandpaper, a crumpled piece of aluminum foil, or steel wool. It doesn't really matter what kind of oil you use. Various people online describe using bike oil, olive oil, WD-40, motor oil, PAM vegetable-oil spray, gun oil, or linseed oil. Ideally the oil should be penetrating (like WD-40) and hydrophobic. Whatever type of oil you use, watch out for damaging your boots, straps, or the plastic parts of the bindings. When you're putting your crampons away in the spring, you can also protect them from rust by coating the areas where the paint has been scraped off with a thin layer of oil, wiping off the excess with a cloth.
The BD article says that the maximum lifespan of crampons is 5 years, and less than that if you use them heavily. I wouldn't take this seriously, since I don't know anybody who replaces them that often. They may be saying this in order to increase sales and avoid lawsuits. I met a guy on Mt. Baldy who had a pair of antique crampons from the late 19th century. They had been carefully taken care of and seemed to be in great shape.
Here is an REI article about crampon maintenance: http://www.rei.com/expertadvice/articles/crampon+care.html
In addition to everything suggested by @Ben Crowell, you will also prolong the life of the crampons if you become very skilled at moving in them. That is, every step or climbing move should be graceful and precise, so that you are not bashing any of the points on anything, ever.
Not only will this be good for your crampons, but it will also be good for your pants, gaiters, ankles, knees, and overall enjoyment of the mountains.
What I believe was left out, is the occasional recoating of exposed steel surfaces with a Teflon coating such as Tuf Cloth or other suitable rust inhibitor. This will aid in protection when the original steel surface is exposed over time.
Stainless steels (I will assume your material is a Cr/Mo alloy here) do eventually rust especially with rough use, wet or possibly acidic/salty environments from say possible salts before trails may greatly amplify this effect.
If the steel shows signs of rust it can be stripped and coated, however if it is left to permeate in layers below pitting may occur, the spikes will eventually fail. This is an extreme however and only a general rule of thumb for rusting steel.