Hi -
I don't think many 6yr olds are very 'independent' but if they have mostly been set clear guidelines, that are realistic for their age and level of experience, and have two parents who love, support and appreciate them (as we were in the fortunate position to be able to do for our DS thirty years ago) then they should have a REASONABLE chance of turning out as decent people, able to make their way through life in a worthwhile way.
But if they have missed out on some of the love and security a child needs, and they don't feel valued, and their efforts and skills aren't appreciated by parents or teachers, then they can be unhappy or depressed, and that will lead to deteriorating behaviour.
I was a (male) TA in an Infant school for ten years. Besides the usual TA duties, I ran a recorder group for Yr2, and coached children on percussion for the Christmas production each year. I also did lunchtime computer activities with Yr6 at our companion Junior school.
I had not been very happy at school as a child myself, so I tried to support children I worked with in a caring, consistent way, expecting reasonable behaviour, but also allowing flexibility for children who were really UNABLE to behave or learn as well as 'normal' children.
Read to, and with, him as much as possible; encourage him in his school work, but understand he may be reluctant to take much 'formal' teaching from you.
Liaise with school, his class teachers, and any Special Needs staff that may be involved with his care. Praise good effort, but be disapproving of rude and unduly silly behaviour.
I always claim children don't really WANT to be naughty, but often outside influences or home circumstances push them into bad behaviour that they can't control.
Encourage activities that he enjoys, drawing, colouring, Lego, sport, music etc. Keep TV and 'tech' to a minimum, but use it creatively when appropriate.
It sounds like you are doing well in somewhat difficult circumstances, and if you have specific problems, PM me if you wish. But mostly, be guided by feedback from school. (Are they aware of all the circumstances? They probably need to be.)