Well the first, obvious & easy thing to do is chuck the jack & the spare wheel out. Frankly I am surprised they let you out on circuit with them there. It is 20+kg you don't need to lug about.
Brakes are largely a compound thing. What you plan will help.
It is a GT-R it will understeer. You can only ever half fix it simply because the weight distribution is rubbish.
If you don't have aftermarket sway bars then get some. As hard as like for the rear, adjustable for the front or just plain soft. I would chuck a rear bar at it then think about the front.
As for the street/track thing - forget it. A track GT-R will run heaps of neg camber at the front . I run 5 degrees. Which is a world away from anything you want on the road. So you need to have a think about what you want. You cannot have a neutral car on the track & a nice one on the road unless you change a bunch of things everytime you go to the track.