Tyre pressures don't make a massive difference, the argument is that you need to raise the pressure for a stiffer sidewall or lower the pressure as the air in the tyre wil get hot and this will expand and increase the pressure anyway. I run mine at 36 as a rule and it seems fine.
As for your suspension issues I think your bang on about it being the springs. GMC efforts are soft as fuck are the reason the car is pitching wildly like that. A set of custom springs will sort that out nicely for you, see the difference bellow in the amount of lean through the long right hander between me and Widders, who was on a set of 50mm springs at that point I believe
Faulkner allow you to specify the poundage of the springs and the length. Mine are 200lb which I think is just right for road use an trackday, a few people have 180lb and some have gone harder with 225 or 250 lb, PGTIs stock hatch 106 has on something crazy like 300lb but that would be a nightmare on the road. Once you've worked out the poundage you want you can decide on the length which will give you the amount it lowers the car by. I can see that your a bit of a style whore Simo so you'll probably want to specify a length shorter than mine as I'm only down 10mm. There is a thread on the SSC where everyone with faulkners put up there specs and how much it dropped the car by, I can't link to it though as the SSC down.
Coilover wise I'm a bit green but you should be able to achieve the same sort of results but keep the car a bit lower down. Like I say though, advice on this is better coming from elsewhere as its not something I've looked into much as I can't see too much advantage in it and the costs are very high, plus I don't want to lower my car to much as I don't like the slammed look that much and it fucks up all the geometry angles.
After springs you may want to look at stiffening up all the bushes and mounts with pug sport kit or similar and increasing the size of your ARB and torsion bars. a 25mm ARB is ideal and about 100quid from pug sport, or you can nab a 24mm one off a 106 GTI or Rallye as they had the thicker ARB as standard for some reason. Torsion bar wise 22mm is ideal for matching up wth 200lb springs and gives a really nice setup. 23mm is availible but may wel be to hard fr road use or 21mm, which matches the 180lb spring very well