Hi Azuil,
Welcome!!!
Your daughter indeed sounds gifted, and even her bad behaviors are 'consistent' with the problems that gifted kids of a certain personality tend towards - not to excuse them, but she will probably need a modified environment (not age lock step with grade) to help curb these tendencies.
Does she read nowadays?
Does she have siblings?
Is homeschooling an option?
Are there 'special schools' for gifted kids within a reasonable driving distance? Is that an option financially?
Yes, I do think testing would be very helpful, provided that a tester-
a) is very familiar with the full range of giftedness in kids
b) can help with some parenting suggestions
c) can help advocate with whatever school she attends
I'm hoping that someone from Chicago can post their favorite tester, or send you a PM, private message. You might also look over past posts in regional, and do a search on 'authority.' (Sadly, this is a frequent topic of conversation here - although certainly not universal)
Love and More Love,
Grinity