I would also sanitize the lava rock by soaking it in bleach water (after a quick wash) for about an hour, then rinsing it well. I also agree with the back tubing (I know you have that planed), also I would light proof the buckets and reservoir if they are not already for the same reason.
However I do disagree about the need for using food grade plastics. Any black irrigation tubing, or vinyl tubing from Lowe's or home depot, like found in the gardening department will be just fine. The reservoir and buckets as well as any other parts don't need to be food grade either. True that some plastics are worse than others. But unless they give off a strong chemical smell, not enough of anything is going to leach into, and build up in the water (especially when you consider
nutrient changes). If your still worried, take a small piece if the plastic in question and place it in some hot water, let it cool down and take a small sip. If it tastes stranger than usual, you may not want to use it. The only time food grade plastics need to be used is if you were trying to grow "certified organic" plants or produce, then yes food grade plastics would be needed. But you would also need to be using "certified organic"
nutrients also. Using food grade plastics is really just a personal choice, mostly made out of paranoia (no offense NorEastFla) in my opinion.
As for the question of which is better flood and drain or drip, really depends on you. Yes both would use a timer, although you could possibly get away with not using one for a
drip system (depending on many factors). I like both of them myself, but both for different reasons. I don't like flood and drain systems for larger setups that would take large amounts of water (nutrient solution) to flood. As someone mentioned nutrients cost money (water's not a problem), that is probably the biggest drawback to a flood and drain.
As for which system is better for, and/or will grow better plants has a lot of factors as well. Especially type of plants and type of
growing medium used, as well as how you put the system together, and where you set the water level. Even the minimum on off times of your timer settings can be a factor.
But going by the pictures of what you are planning, I would suggest a combination flood and drain/water culture system. Mainly because of the five gallon buckets and the pots for the plants that I assume will be sitting in holes in the lids. As the roots grow down into the five gallon buckets from the pots, they will dry out if not able to get moisture in-between flooding. So I would set it up as a flood and drain, but with an adjustable water level when it drains. That way some of the roots that hang down will be able to get water all the time. I would make that level adjustable so as the roots continue to grow down they wont be completely submerged.
As I mentioned even your timer settings can be a factor. Here is an example. If your timer is capable of 15 min on and 15 off settings around the clock, the roots that are hanging down (in air) probably wont dry out that much in 15 minutes in the setup I explained. But if the minimum on off time is 30 minutes, I would be concerned about them drying out between flooding. Either way I would probably still want to make the overflow adjustable. I would still want some water at the bottom the buckets for moisture and humidity. Also if you decide to go with a drip system I would still want some water at the bottom, again an adjustable overflow.
I attached a few pictures of a easy adjustable overflow for buckets like that I have done. It's nothing more than a through hole (also known as a bulkhead fitting) with a piece of vinyl tubing stuck in it that regulates the water level at the bottom. To adjust the height, just change the height of the vinyl tubing. In case you want the water level high and the vinyl tubing keeps bending down, stick it inside a piece of PVC tubing that will keep it straight. You can also just use a piece of PVC tubing in the first place (instead of the vinyl tubing) as long as it tightly fits inside through hole. Or gluing an PVC adapter to the through hole that the PVC tube fits in, so you can easily change the tube. I can think of lots of ways to make it adjustable using the through hole at the bottom. I got those through holes at home depot next to the electrical conduit.