I was very daunted by driving when I started, especially when I had to drive a really old clunky low-power diesel Mercedes with my overly critical dad and big brother yelling at me.
Didn't take long for it to become second nature, and I actually drove taxis for several years.
It literally just takes a while for your brain to get accustomed to everything. Yes, youre right in that a manual can be a distraction from learning traffic, but it really doesn't take long for the shifting to become very natural, and then you can pay attention to the traffic.
As for actual tips on the clutch, it's not really as much a matter of millimeters, as it is about the balance between throttle and clutch. Don't be afraid to give it a bit more gas to makeita sure you won't stop the car even if you let go of the clutch slower. Yes it will sound a bit like a student driver, revving the car "needlessly", but you are a student driver and it isn't needless revving, because you're still learning the feel for the clutch.
It also depend a lot on the make and model of the car the school has. It's a bit of a personal preference, but clutches are really different in different cars. You might really like a 90's Mercedes with a reeaally long clutch, but that too takes getting a bit used to. I remember the cars my car school had were fairly new Ford's, and they had really short clutches. Annoyed me as well.
Honestly took me less than a year of driving and I knew how to shift without a clutch. Some really old 70's sand lorries my dad used to drive didn't have clutches, so he taught me how to (in some cars it works some don't really like it, but basically you could do it in all manual cars, without damaging the gearbox if you do it properly); when accelerating, just as you let go of the throttle, like 0.5sec after that there's a short period where you can just pull the gear to neutral without any resistance. That's easy. Shifting into gear from neutral without clutch isn't as easy, but in some cars, not much of a challenge. You need to rev the engine to match the rpm, but like with a short press of the gas pedal, which revs the rpm higher than it needs to, then when the rpm is coming down there's a window where you can shift into gear really easily.
I ramble, you don't need to thin about that.
I would suggest, if possible (idk if you have a learners permit or smth and can drive under parent supervision), to practice driving on highways and country roads for less traffic to learn the car better. If not, maybe ask your teacher on the next lesson if you could do that.
And if not, if you have to drive in the city, the most important is just to remember that the panic and rudeness you feel isn't something which should make you hurry. People will be annoyed, but you have a right to annoy them, as you're learning. It's a shit thing, being a learning driver, but once you get your licence and get to take the car on a long drive yourself on some chill roads, you might actually enjoy the driving because the stress won't be there. And then learning will become easier as well, when your heart isn't beating through your chest with someone watching over you.
Just keep at it. You know what to do. Now it's just a bit of repetition.
Edit oh and ask anything if you like, 3rd gen taxi driver, drove since 2007 (not driving currently). No stupid questions exist. So anything at all, go ahead.