i have had bad jealousy issues with my partner (i'm the jelous and anxious one) and to be honest our situation sounds very simular!
First off, about
the social thing - you might find that he does enjoy socialising with your friends, but he just gets too worked up before hand to be able to even get to the fun part - that's very much what i do. Try having a couple of friends over to your house perhaps - it might be the going out part that makes him worried. And then when he gets used to those people and forms a good friendship, maybe start to introduce small outings. I think the trick with these outings and gatherings away from the home is to keep then short and small at first and slowly building it up. Confidence is a very delicate thing and needs to be built on very gradually.
Also, with my partner, i find they are so so so excited and so desperetly want me to be apart of their plans that they forget that i might not feel up to it - so talk to him - just say you know "i'm doing this with these people, i'd love for you to come, but i understand if you want to stay in tonight" and then if he doesn't come, no need to worry, just come home and tell him all about
it, and he may well fancy it next time.
As for the jealousy issue - its a tough one and one myself and my partner haven't yet been able to competely deal with. I think the most important thing is to be able to talk with each other and it sounds like you are able to do that. Try and keep in mind that you are supposed to be on each others side and you are there to make each other's lives better and to support each other. Try and think about
any patterns that form - is it with certain people that he gets jelouse? Could there be a reason for this? Does the jealousy occur when he's feeling particularly low or anxcious - could it be a reflection of his low self esteem. A very important thing to bare in mind - that as hard as it is when he is jealous - he gets that way because he loves you - but this doesn't give him an excuse to behave that way if its unfounded and is making you feel bad. Let him know how you feel.
this is a great helpful place.