I'm sure others will chime in too, but I will respond to some of your questions. Others on this board have helped me to understand that giftedness looks different in each kid, and it isn't so easy to say that GAI =1xx, and so kid will be doing y. We homeschool, so I don't know for sure, but I actually think my EG/PG daughter would be doing very nicely at the prestigious and challenging private schools in my area, but I don't think she would be hitting anything out of the park. She is introverted, cautious, and perfectionistic. She is also slow and cautious. My son, on the other hand, who hasn't been tested but is likely also HG+, might have a different set of issues. He is high-energy, out of the box, gotta question authority in any form, school is boring so let me get it over with as fast as possible. Bright? Extremely. Creative? Hell, yeah. A good student? Again, I don't know for sure, but I imagine not so much. Bottom line -- HG+ does not necessarily equal being a super high achiever.
As for achievement tests, if the school won't administer, you could always have him take the EXPLORE next year.
Good luck!