Tatertot, I have a daughter (now almost 11) who is “highly gifted” (that’s the school system’s choice of words, not mine), and it’s an interesting experience, to say the least. It sounds like you’re moving along nicely!
As far as testing goes, at the kindergarten level it’s wise to let the teacher take the lead. The tests are pretty time-intensive and take a fair amount of concentration, so it’s hard for a kindergartner to sit through them without losing interest. As long as you’re happy with the teacher and the school right now, you can afford to wait and have him tested later. Eventually, it IS nice to have a piece of paper that says, essentially, “my kid really honestly needs some extra attention - I’m not just a proud parent.” I suggested to my daughter’s first teacher that she might need to be tested and she basically patted me on the head and said, “They’re ALL special.” Six weeks later, she came to me and said she thought we should have the child referred for testing. Decent teachers notice these things after a while. 
With gifted kids, I think the most important things are to make sure they’re not bored/frustrated with their schoolwork and, especially at this age, that they’re learning how to get along with others. Kids who have a well-developed intellect often have a corresponding deficit in social skills (my own precious darling became notorious as a vicious hugger and had to be restrained from showing physical affection until she learned not to tackle other children). I think it’s worthwhile to concentrate on that side of their learning during their first years in school, so the kindergarten he’s in sounds ideal to me. Skipping grades is also often a bad idea; the best thing to do is to find classes where he can do work at his level among children of his age (difficult, I know, but possible). It’s a fun ride!
Here are a few links that might be helpful:
Support Society for Children of High Intelligence
World Council for Gifted and Talented Children
The National Foundation for Gifted and Creative Children
If you’d ever like to talk about the joys and challenges of dealing with “special” kids, please email me.