My 2yr8 mth old son shakes his hand whenever anxious,

Hello. My name is ***** ***** I would like to address your concerns. My background is assessment, diagnosis, treatment, & case management of services for individuals with autism, pervasive developmental disorders, and other issues related to learning, behavior, & development.
Your description of behaviors (i.e. hand shaking (flapping) when nervous or over excited, walking lines in repetitive manner while using odd angle to visually inspect, preoccupation with wheels & spinning objects, memorization & reciting, and echoing things that have been said to him) is consistent with behaviors that a child with autism may display. It also appears that if does have an autism spectrum disorder it is more high functioning given your description of his intellect. I can not give you a diagnosis in this format, however, I can recommend that you seek a comprehensive evaluation from a neuropsychologist, psychologist, or psychiatrist who has much experience with autism spectrum disorders. If they are good at what they do, they should be able to evaluate, diagnose, & recommend a course of treatment.
In order to find autism experts, I recommend looking for local, national, & global autism awareness and support groups. Often times, these parents have gone what you are going through and can be a resource for both information and support as your son goes through the evaluation process.
I hope that this has been helpful and I wish you the best as you seek additional information to support your son's development.

Hello,
Cardinal features of autism are lack of communication skill, (not verbal) fixating to repetitive acts, not capable of imaginative play - child seems alone, detached and
withdrawn.
( lack of symbolic and imaginative play not purposeful activity but often more a fiddling with objects using toys/ objects not as their intended functions)
Most autistic kids are not capable of anxiety. Movements they have may be stereotype but are not in response to any external pressure due to anxiety.
Your child does not have these features. Please visit link to compare symptoms with classical ones mentioned : http://reachingautism.org/diagnosis.html
He does however need in person evaluation by an expert in child behavior.