what is the air button and valve for in the trunk of my '03 ssei?

The air button activates the compressor and fills the rear shocks. If you push it after a little while you'll notice the back end of your car is sitting higher. The valve is an air valve and I do believe is meant to be used with the hose that'* in the accessory kit for airing up the spare tire or something

Yep, the valve is there to fill tires.

I noticed that if I raise my car using that button, that it will after a while fall to its normal height. What then is the purpose?

the shocks on the 00+ double as air tanks to hold air to air things up. if the air is not used the relief vavle will open and let the air out to bring it back to ride height

At what point does the pump stop compressing air?

it stops after it'* 15 min run time. as far as i have observed the excess air just bleeds off thought the vavles in the shocks

I've heard it done to gain more traction at the track by pushing the center of gravity farther toward the front of the car. How does one accomplish that?

start car, push air button and the shocks will fill up. i do it in the staging lanes while i'm waiting to take my next run.

learn something new everyday!
i used it few times,didnt know/notice that itll lift up

thanks for the info!