I've read that you can adjust the pitch by altering the key heights. What is the process?

It is true you can lower pitch by reducing key height, but almost always at the expense of a full voice for that note. If you raise the key height above 30 percent of the tone hole diameter it will make no difference whatsoever. This is a fool's game, and not the way to tune a saxophone.

Are there charts or references available which show the correct heights for various saxophones? I have seen gauges offered for sale by repair tool makers, but would suspect they offer only generic and not make/model specific settings.

I'm not aware of any reference material, and probably wouldn't use it if it were. You have to set the key heights to make the specific horn you are working on play well, and if you can't figure that out, you need to hire somebody who does to do the job.

If you will experiment a bit with the openings, I think you will find that the horn speaks much easier and more freely if you have the keys pretty open. Too closed and it will be stuffy and dead sounding.