Lets say I don't aerate, and I have a plant with roots half submerged in the water. What happens? Do the roots below the water line die?
No the roots won't die. (That's the principle on which the Kratky method is based on btw.).
The problem though is that you will run into bacteria problems with un-aerated standing water if you're not super careful about hygiene.
So while it works I'd suggest aerating at least when you're starting out.
I have an aquarium aerator, can I safely assume that it adequately aerates a volume of water of the same size as the aquarium its intended for, but for plants instead of fish?
As long as you have enough bubbles (or plume) on the water surface it's sufficient. If your fish don't die plants will survive too. (Aerators don't put air directly into the water. They just break the water's surface tension so the atmospheric oxygen can pass into the water. So as long as the surface tension gets broken it's all right).
Is aerating an issue if you only have the pump kick on once in a while, alternating between flooding the plants and letting them drain thoroughly, giving the plants a chance to breathe?
Don't know. Never tried. But I guess that would work. You would just need to make sure that the roots don't dry out.
Are there any lesser known considerations you wish you knew about before you started?
Yes, make the water reservoir(s) as easily accessible as possible. When plants get bigger and you want to change the water you're in a world of hurt when you don't have easy access to the water reservoirs. (Taking EC/PH readings, changing the water, etc. becomes a huge pain).
Source: my green house where I put tomato plants into deep water culture (each plant gets its own bucket). It takes considerable work to pump out water out of every bucket and fill new in. I hate it that much that my plants don't get adequate care because I tend to defer water changes as much as possible. Next year I will do something with one easily accessible central reservoir which then circulates the water to the smaller plant buckets.
Also: Water chemistry is a bitch. Buffering effects will give you a lot of head ache until you figure out how to deal with them and prevent PH drift. Get reverse osmosis or rain water if you can. If you can't then don't forget that there are hard water versions of most hydro fertilisers available which you might need to use depending on your water's quality. (You most likely will have to use them if you have average to hard water).
Also2: You will find a lot (and I mean A LOT) info related to marihuana when you google for hydro related topics. Don't listen to the advice there. It's mostly stoner cargo cult without any scientific background. Use info for commercial growers instead. Those are the people who grow tomatoes in hydroponics and know their shit.
Also3: Aquarist blogs and forums are a great source to learn about water and water chemistry for people who don't have a degree in chemistry.