Hi Cheapo, nice to see so many new faces around here recently!
ER has taught you the basics well and you've done good to listen well...
1. You're describing three principles in one: the first is to use every inch of the track to make the corner as shallow as possible to carry as much speed through it as you can. Ideally, you should turn in from the outermost point of the track, clip the inside as far inside as possible, and the speed and your acceleration should carry the car to the outside edge again. Have they told you about early/late apex and how it's sometimes better/faster to compromise the corner?
The second is not to coast - usually every moment you're completely both off the brakes and off the throttle is one you're not getting the most out of the car. Now, it wouldn't be racing if there wasn't also an exception to this rule: the corner, the car and/or the setup may dictate a degree of coasting - you'll find out when to with experience.
Lastly, smoothness... A racing idiom is "smooth is fast". It is important to note though that this doesn't necessarily describes the input, but the movement of the car. An overpowered, heavy cow on too narrow tyres (like V8 Supercars or the NASCAR-series) will need to be driven gingerly (the orange your instructors were speaking of), while a small and light car (maybe even FWD) may need to be manhandled to really be fast. To go off into a tangent (a small glimpse into the rabbit hole that are tyres), tyres have the most traction when they are just slipping a little bit (called slip angle for lateral movement and slip ratio for longitudinal) - it is a knife edge balancing act. One side means being slow, the other may mean your race is over (in real life that means damage you have to repair/pay, but also injury).
2. There are maybe a handful of drivers better than modern ABS systems, so using it if you're allowed is certainly advantageous. Keep in mind though, as the tyres are locking for instants, it can have an impact on tyre wear... Also, full on the brakes may prevent you from turning in efficiently, as all the traction is used up for decelerating the car...
3. This comes back to smoothness, but you already have the right idea... If you're going down one gear, your engine is running much lower RPMs than the driveline, so clutching in too fast sends a jolt through the driveline which upsets the car balance and maybe even causes tyre lockup. Now, any of the three pedals can prevent this: clutch in slower, but that causes slip therefore higher wear and temperature on the clutch, let off the brakes slightly, but that means you compromise your braking possibly for longer than the jolt lasts, or you could speed up the engine in the time you're shifting, which means you need to learn the art of operating three pedals with two feet, or heel toe for short.
Heel toe doesn't mean that you use exactly your heels and your toes to push the pedals, any contortion of your body that lets you push the throttle while still keeping the same amount of braking is fine.
4. Trail braking is just keeping on the brake while you're already turning into the corner. It's just a technique to keep the balance where you want it. Some cars (or better: some setups) don't like trail braking, while others almost require it. For example: if your car already pivots over the front wheels if you only think about turning you'll probably only swap ends with trail braking, whereas if it understeers heavily on turn in, trail braking will help alleviate that.
Brake balance is also an important factor: a balance too much forward will only overpower the front tyres and you'll only plough straight on, too much to the rear and you'll also just swap ends...
Phew, long text - seems I've written even more than you, hopefully you find these long-winded answers useful...