Any good relationship is built on trust whether they're an aspie or not, but this develops over time and you have only just started dating.
He sounds insecure, perhaps he's even been cheated on in the past. I don't think most men whether aspie or not would truly like their girlfriend being friends with someone they've slept with in the past, especially if they haven't being going out long and I think most would feel somewhat insecure by this, however a non aspie (NT) maybe more discrete about it knowing that it's probably best to show that everything's fine as otherwise it shows distrust and perhaps even jealousy. An aspie however may not know how to be discrete and may instead show their true feelings whether it's appropriate or not.
It does sounds like he likes you a lot however as he is worried about losing you, the problem is when people over act trying to ensure they don't lose someone too much, it ultimately causes them to lose them. I'm not saying you should break ties with your friend and he has to realise that you have a right to have friends, but if you truly like your new boyfriend, then you would most likely need to keep giving him reassurance that he won't lose you and that you're only interested in him, but the problem is with aspies is this reassurance will keep wearing off and you will most probably need to keep repeating it to keep him feeling secure again. Once you've been in a relationship for sometime however you should both learn to trust each other, it should happen eventually even with an insecure aspie, although it could take a lot longer with a lot more patience (if trust doesn't eventually develop between 2 people in a relationship over time, then sadly they're probably not right for each-other as trust is vital to a successful loving relationship).
Aspies may not always fully understand or think about other people's feelings in an emotional situation and may instead just focus on doing what they consider to be making something right instead. He probably feels that the child and yourself are being mistreated by your ex because you're not receiving child support when lawfully you should be (it's unlikely he's in any way interested in the money himself).
Aspies can also obsess over things way too much and sometimes it could even be over something that other people wouldn't expect, certain things could worry him and keep going round and round in his head unresolved. He may well go on and on about something like this, often repeating himself without considering or fully understanding that you probably don't feel comfortable with it, it won't be his intention to upset you and to him he probably feels like he's just trying to help and stand up for your rights.
You need to talk to him and explain your feelings slowly to him, but you need to be patient and understanding. He may need to be spoken to or reminded on a few occasions to stop him from obsessing over it.
It's great that you're here learning about aspies, you may also wish to read up on common traits of
Asperger Syndrome if you haven't already. It is definitely possible for a non aspie and aspie to have a long loving relationship, it will often take more patience and understanding, but it can be well worth it in the end and aspies are often the most trustworthy and loyal companions of all. Sometimes he may do or say things that some may consider inappropriate or maybe even offensive, but usually that won't be intentional and if you have patience and understanding you then will learn not to be annoyed or offended by it, E.g. it's just the way he is, and you may even learn to love some of his traits.
Aspies can still learn to control traits better however and over time you will hopefully be in a position where you can work together to achieve this (I'm now 48 and I've learnt to see a lot of my traits and control them better, although I'm far from perfect and can still say or do the wrong things sometimes and without realising offend people, also relationships never worked well for myself, although there are many aspies who have been luckier and are in happy long term loving relationships).
Has he got a special interest or interests, something that he often obsesses over and spends a lot of his time doing? Not all aspies have a special interest, but many do and they can often obsess over it. If he does have a special interest or interests, or if he later finds one, please don't be offended if he seems to spend more time on that than with yourself, for instance in future he could tell you that he'll be up in 10 minutes as he's still working on it when you call him to bed, lol! I know I've done this to someone when I was younger and I've later realised, it wouldn't mean he loves you any less however. If you could take a least a little interest in any special interests he would most likely appreciate it however, you don't have to love or even particularly like his special interest, but just being there occasionally to view and acknowledge his achievements would usually be greatly appreciated.
Finally don't listen to your friends who probably haven't took the time to learn about Asperger Syndrome, do what YOU want to do. I wish you the very best of luck and please feel free to keep us updated.
Edit:
I would make a clean break.
He sounds like a control freak, and if this is happening so soon into the relationship, I dread to think what it might escalate to.
Aspergers or not, it is not an excuse to manipulate you into making decisions about your life that are not anything to do with him.
From the limited information given and known at this time I feel it's probably too early to make a definite assumption like that (my opinion). I most definitely understand why you say this however, alarm bells would be ringing for a lot of people and you may end up being right. If he confidently starts giving orders and won't ever listen no matter what or worse still gets aggressive, E.g. he orders, "you will stop seeing your friend" without any possible compromise, then I would agree with you, but I think he deserves to be given a chance at least and Asperger Syndrome probably has a play in the way he is acting and/or coming across.
Note: Also people should be a little careful not to misinterpret someone with Asperger Syndrome as being aggressive, sometimes they're not really being aggressive even when some people think they are coming across that way.
