how would rake change affect the car?

You need to measure at the factory points with driven tire pressures. But be mindful that some race tire/car combos have a significant 6-8psi hot to cold tire pressure difference and might be run under or over the recommended hot pressure for that tire as run can effect the real ride height. For example, the Hoosier Dot tire is fastest on vettes way under Hoosier hot pressure recs while Hoosier recommended 4psi higher than the recommended hot pressure while running at Daytona speedway for the SCCA Runoffs. That would be a significant ride height delta.

In general lowering the rear tends to add grip in the rear until it doesn't and raising the front makes it looser in the front. Overall you have put more grip in the rear and your brake transfer is slowed. I don't know how much aero you can really put in a streatcar since streetcars with full wings and spliters are not really significant like aero in a purpose built racecar. So how the areo balance changes I don't know but you prefer aero understeer for safety while the lowest laptime cars tend toward mechanical oversteer

 

I would think there are many variables. In my own case I have to be ready to race with 1 tire scrub warmup lap then hit it hard into the 1st turn. But that is wheel to wheel racing. A DE person tends to build speed over a few laps so I would expect a slower tire warm up. Track surface and driver style is yet another factor. I'm pretty certain ,but do not really know, that my tires have not fully warmed up after my 1st warmup lap to a green flag race start. So I always know which direction turn 1 will be in and I warm that side the most during the warmup lap. I suspect that by lap 3 you are hot and stay that way unless you over heat fighting someone in a close battle or mounting a challenge. During a race you can feel your tires loose grip and regain grip as you control the temps by your on track aggressiveness. You need to find a sweet spot and use your pyrometer in the hot pit for a good starting point.

 

You sort of are getting normal occurrence. in my 3500lb corvette racecar my tires grow from cold to hot 8psi on a dot-R race tire. I am not sure what a street tire will do. But from cold to hot getting a 4psi change is reasonable and you expect the tire to grow. Just coming into the hot pit and dropping the pressure you immediately go out and the tire will grow because you put heat in tire when you drive. You can keep a tire at 2.1 if you come into hotpit and check then go back out and come in and check again. You can cool the tire for 30 mins or more then go out and go to hotpit and check and still be at 2.1 because your tire grew back up as it warmed up. When you go home the tire will cool from that 2.1 and maybe 1.9 cold. This is all potentially normal.

If you want to find the best tire pressure you need to use a tire pyrometer but street tires are very thick made to last 50,000miles. A race tire is good for only 1 race and sometimes does not survive the race if pushed too hard. So I question the value of using a pyrometer on street tires. You also need to push near the limit of the tire grip at that pressure to get proper tire temp readings with the pyrometer. If you are driving 5/10ths and not heading the trier up much during the session you are not going to get accurate tire temps and thus know the right hot pressure of the tire.