Welcome!
Your child appears to have accomplished enormous growth and high achievement with your support. You know your child best. As discussed often on the forums, there is a fine line between supporting a child's growth and abilities by setting high, reasonable expectations -vs- pushing a child, which in many cases may not be sustainable and/or may lead to social, emotional and/or trust issues. (Search for posts on hothousing or tiger parenting.)
I am confused on my child and the question of Is he or isn't he gifted?????
Formally identifying a child as "gifted" most often occurs through a process of IQ testing.
My problem is that my son did not just start reading on his own or doing 3 digit multiplication on his own.
Gifted children do not match all of the common early milestones. However there is often a complexity of thought and ideas, evidence of making connections between concepts, and a creative/innovative/novel aspect to their questions and answers. This may be different than rote recitation of facts.
He has worked really hard to get where he is today... he did not just magically get there on his own but he and I have worked together on his reading and math.
Even children who meet early milestones do not do so magically but are typically thought to have keen awareness, strong powers of observation, internally motivating levels of curiosity, and/or focus which fuels their ability to learn from casual exposure to concepts from others in their environment (indirect instruction). For example, experiencing a parent reading to them, they may begin to match sounds to words on the page, without explicit direct instruction.
Does every child have the potential if they applied themselves to test higher then their current grade level?
Unfortunately, no. For example, a child with an IQ of 70 may not perform like a child with an IQ of 130. However, current understanding is that a child with an IQ of 70 (nature) may have strengths and interests which can be built upon... and may be supported (nurture) to learn more than children with an IQ of 70 typically learned in the past.
Can I expect other children to catch up with him in 3rd grade with their math and reading abilities?
This depends upon the child and the environment (nature and nurture). Although external systems may frequently make comparisons among individuals, internalizing a pattern of comparison with others may become toxic. Some may say it is better to focus on being the best self one can be, rather than focusing on being the best in a certain group, when measured by specific criteria.
I think 1 hour of school work, outside of school/summer in place of his eyes glued to a computer/TV is not too much to ask for in a child. Am I wrong?
This may depend upon whether the learning is fun and engaging, especially for a young child. Kids may benefit from being encouraged to listen to their internal voice which expresses an interest in a topic or area of study, and receiving support to find resources to feed that interest (which reinforces being internally motivated), rather than having areas of study prescribed (which may eventually lead to a child being externally motivated and possibly feeling quite under-confident in choosing his/her own path and persisting on his/her journey).
Also, do not underestimate the value of physical activity and imaginative play.
The work of Carol Dweck, and her book
mindset may be of interest.
