Estonia; what's the climate like? Lettuce doesn't like heat, just in case it's hot there.

Lettuce may be better grown on a ramp (floating in nutrient). Also, growing seedlings in dirt, you're likely going to pick up some soil-borne critters and maybe microbes, which you really don't want. Try to transplant your seedlings as little as possible. Once you have more experience, it's not a big deal, but for a beginner, it's way easier to stick to one method at a time.

It would be better to start your seeds hydroponically and when the plants are about three inches tall, suspend their main stems above the NFT trough so the small white roots dangle in the trickle of nutrient. I would use perlite to start the seedlings; it will shake off the roots easily when it's time to transplant to the NFT system.

Suspending the little plants into the NFT system can be done any number of ways. I've done this and used small wads of fiberglass wall insulation to hold the stems while the seedlings grow. At harvest time, I merely cut the plant at the fiberglass and throw the roots and all into the compost heap.

If you're going to use NFT, you don't need the little pots full of expanded clay balls. Just suspend the seedlings from the top frame of the NFT trough.

If this is your first try at hydroponics, my advise would be to start with something a little simpler like growing using static culture in perlite or using a ramp, which is a raft that holds the seedlings on a lake of nutrient 24/7.

Well one thing we don't have enough in Estonia is heat :D
I would still like to start with the NFT because the lettuce is just an experimentation. If everything goes right I would be starting to grow other plants.
I have noticed that when the dirt around the lettuce roots is dry enough it would come off real easy and the small stuff that still is on the roots, could just be washed away.
I have also thought to make a system that is similar with the NFT but uses drip irrigation. Water would still flow on the bottom of the pipe but it would all come separately to every plant from the top. I hope you understand what I mean. What do you think of this plan? Or would the usual NFT still better?

Sounds like you're going through with it.

Do yourself a favor: If you're going to start the seedlings in dirt, wash the roots very carefully in tepid water. If you try to shake the dirt off dry roots, you'll damage them.

There are very tiny almost hair-sized roots that need to remain in tact for the seedling to stand any chance of transplanting from soil to NTF or whatever method you finally choose.

As far as exactly which method to use, I'm a firm believer in "KISS" (Keep It Simple, Son).

I have also thought to make a system that is similar with the NFT but uses drip irrigation. Water would still flow on the bottom of the pipe but it would all come separately to every plant from the top. I hope you understand what I mean. What do you think of this plan?

Drip irrigation (usually with a medium such as sand or perlite, to retain moisture) with subsurface recovery like you described is one of the most, if not the most, widespread commercial methods of hydroponics.

Be careful with letting the roots of your plants dry out. That's a good way to kill a plant.

One idea that might work, depending on how your NFT system is built, would be to raise the point at which water drains out of the gutter to a level that would cause it to fill up with water a bit. That would make it easier for the plants to reach the water and not dry out, then when the roots are long enough you can gradually lower the drain until it becomes NFT.