fjzh,
I glanced through the provider website that you linked. It looks like they do a range of assessments, from basic cognitive out to comprehensive neuropsychs, and they have contracts with allied providers (OT, speech, OG/Wilson/Lindamood-Bell reading remediation). I don't know anything directly about them, beyond their website, but the people I can find info on seem pretty reputable.
Whether the professionals who would see your child would be a good fit will depend also on the rapport you and your child feel with them at the intake. Wherever you decide to go, do not be afraid to back out of it, even after beginning the process (preferably before actual testing has commenced, though), if you or your child feel uncomfortable with them. Even excellent clinicians sometimes have certain clients that just don't connect, for no apparent reason. That's not a knock on either clinician or client--merely a poor fit. And a good reason to give your child a chance to meet them before beginning testing.
As to what kind of eval: Since your questions are really more 2e in nature than straight gifted or straight LD, I think you might be looking more for a neuropsych than a basic psychoed eval, which -could- include everything you need, but might not. More important than the name of the eval you are requesting is describing your concerns, and what you want to gain out of the eval.
E.g.,
1. DC has such and such concerning presentations (emotional dysregulation, sensory defensiveness, perfectionism);
2. we suspect these things may be factors (giftedness, anxiety,...),
3. we would like to gain this information (cognitive level, presence/absence of 2e), and
4. recommendations for these settings and functions (school, home, peer relations), including
---a. remediation,
---b. accommodations, and
---c. realistic future planning.
