I didn't vote because it really depends, your ex was a huge part of your life and if you ended on good terms then thats great! However it really depends on why you broke up and how long you guys dated, personally I don't see a point in hanging out with an ex unless you want to rekindle something, I've never after a break up began to be friends with an ex because I've never thought of them as a friend in the first place, there was always romance, and love and it's just really not that possible to go back into thinking of them in a platonic way, they will always be an ex. Maybe your girlfriend doesn't get why you two are friends, and or would want to hang out in the first place, ask her why and see her reasoning it could be totally valid. If your ex is in the same friend group then thats not really avoidable, and unfair.
That all depends. How fresh is the breakup? Has that ex moved on as well? Does she act/talk flirtatiously to you that would make your girlfriend feel uncomfortable?
But, no matter what, you should respect your girlfriends, not laugh in her face, I think it's a legitimate concern of hers (at least in her mind). Try to sit down and explain to her that you are only wanting to be friends (I hope that is the case here) and that you are in a relationship with her because you want HER not your EX. If she still won't budge, try asking if she would be comfortable hanging out together in a group of friends or not. (Probably not but you never know).
But ask yourself, is the relationship you have with your girlfriend more important to you than the contact you have with your ex? If you can't respect your girlfriends wishes to not hang out with your ex, you should evaluate whether you should be in that relationship or not.
It's understandable that she has some concerns. You have every right to insist on spending time with your ex if that's important to you, but the least you can do is talk to her about her feelings and be reassuring. It's very normal to feel insecure about that type of thing. She's only human.
I don't believe in trying to control what my partner does so yes, I would allow him. However I would make it clear that I'm not comformable with the idea at all and that he's free to to whatever. I just won't be happy about it if he then decides to go. If he goes anyway, that'll tell me what's more important to him. My feelings or hanging with the ex.