Hi, Having diabetes should not stop a child from leading a normal life, but it does take some management and understanding of the condition. I don't see why your boyfriends son should not be allowed to stay with him, as long as both you and your partner have a full understanding of the condition and know how to adminsiter injections, treat hypos and work out insulin doses and corrections.
Once you have a full understanding of this then it seems perfectly reasonable for your boyfriends son to stay with you, although I can totally understand his ex having difficulty in letting go.
I myself have 2 children with type 1 diabetes aged 5 & 7. My 7 year old has only just started staying with her Grandma as we felt she was ready to as she has recently started to do her own injections. We told Grandma her set routine, but Grandma called us at meal times and told us what she was having and we worked out the insulin dose and told her how much our daughter needed. We then got her to do blood tests every 2 hours and phone us with the readings so we could make sure she wasn't going too high or too low. She also phoned us just before bed and first thing in the morning.
May be you could have this arrangement with your partners ex, so she is still in control and knows what exactly is going on. It won't stop you going out and doing stuff and you can keep her updated at all times.
Ask her what her concerns are and you can then deal with them in the way she wishes. With regard to the injections I think it is hard to put an age on being able. My daughter turned 7 in Dec and started in Jan. Since she did her first one she has never wanted us to do it. I feel however that she is quite young, but I know of other children both younger who can do it and some children who are older who still have a parent doing it. May be you could speak with his mother about arranging to see his DN so you can have some training in administering the injections (which are quite straightforward).
When mine were first diagnosed I practised on myself so I knew how it felt.
I hope this helps and that you can look forward to more than a few hours with your partners son. Leggott