When doing NFT in pipes, figuring the bed size doesn't really do much for you since you get very little filtration from the NFT pipes.
You will need some sort of filter to take care of bio-filtration and solids filtration before you send your water to your pipes or you will have problems with gunk building up in the pipes and causing a stinking mess and stunting your plants.
The plants get the nutrient that is in solution in the water. If your solids and fish water go into some form of filtration and the bacteria do the conversions in the filter and the water that comes out has the nutrient in it.
Yes I think a media filled grow bed is a great way to filter water before sending it to you NFT pipes. There is an option that you could set up a small sump tank below your grow bed and have the water drain from the grow bed into that "clean water sump" and have a small pump in there to feed your NFT pipes which is an option if raising you bed or lowering the nft won't work well. You would probably need an overflow from the "clean water sump" back to the main fish tank in case your NFT pump was too small to keep up and of course you would need to make sure the "clean water sump" had enough water or enough flow into it to keep up with the small NFT pump so it didn't run dry.
Hope this helps.
You need something to start the cycling, so if you don't have fish you may need some chicken manure or other ammonia-producer. Then the bacteria in your system will kick in and create the nitrates for your plants. The nutrients are in the water, so filtering solids is simply to keep things flowing smoothly.
We have a flood & drain system, but will be adding some floating beds. It all works on the same principles, though. You have to have ammonia producers and you have to have bacteria and a place for it to grow.
We have a separate bio filter that our water flows through, with lava rock and filter media. That way if anything happens to our hydroton grow bed, we still have bacteria in the system, and if anything happens to our filter, the bacteria is still in the hydroton. We also keep some rocks in the bottom of our tank. That gives at least three good places for the bacteria to grow. I've worked in technology long enough to know that a backup is essential, and two backups are that much better.