No not all men who move fast are abusive.
But there are multiple red flags here. Remember that a red flag in itself isn't a guarantee that this is an abusive person, it's just a warning sign that they might be. It's about looking at the pattern as a whole and not just taking each red flag at face value. For example I'll show you the ones you've mentioned. (Sorry, this ended up really really long!)
You feel an urge to run away
Red potential: It's a genuine, instinctive response because something isn't quite right in the relationship that you can't put your finger on.
Benign potential: You have unfounded fears based on previous experience.
He's full on/moving the relationship too fast
Red potential: Abusers tend to be insecure and they want to rush you through in a whirlwind because they are worried you will see the real them and run away or because they are quite emotionally immature themselves and live relationships out like a teenager, or because it's been successful to them in the past - charming you off your feet, putting them in control.
Benign potential: He's just excited/perhaps a little immature, but not in a dangerous way, he's feeling swept up in everything and getting carried away.
He's quite clearly anxious about losing you/you cheating
Red potential: This is a big one, TBH. Abuse is motivated by the "need" to control another person. (I put need into quotes because nobody needs to control anybody else, but sometimes we think that we do.) One cause could be difficulty trusting. The fact that he's made it clear already that he's anxious about you cheating or the fact that you could choose to walk away at any moment is concerning, especially so the more often or the more strongly he expresses it because it means that he's not comfortable with the fact you're an independent person who can choose to act outside of ways that he's in control of. The problem is that relationships aren't a game of master and puppet, and to have a relationship he needs to accept that you're a separate person from him and you might well choose to cheat or leave. Cheating is, of course, a horrible thing to do to someone in any case but him being SO anxious about you leaving, after a month (or earlier) is less of a red flag and more of a wailing beacon that he's got some issues which he needs to seriously work on before getting into a relationship. It's extremely likely that he's either going to turn out to be an abuser or that he's just going to drive you crazy with his insecurity and general downward spiral on everything.
Benign potential: If I've misunderstood and he's not coming across so strongly with this and you're making assumptions, then it could be that somebody in this situation feels insecure but is actually fairly in control of their own feelings and although they are anxious about it they are keeping it in check and won't let it affect the relationship.
Note: I want to make it clear, though, that it's not normal or healthy for somebody to be expressing worry about a new partner leaving them after a very short time together. That is a massive sign that they are seeing relationships as more like something where you "catch" a person (to use a metaphor, a fish, a pokemon) and you have to do/say all the right things to make them stay and if they stay then you've "won" and everything is fine, but that if you're unlucky or you do something wrong then they will escape again and you'll lose them. This is unhelpful, because how healthy relationships actually work is that two people express interest in each other and then they spend some time together to see whether they work as a couple and how they fit together. During this initial time one or both of them might realise that the fit isn't really right and they aren't a good match for each other and so the relationship will end, and while that can be hurtful and upsetting, especially if you felt like there was potential, it's not a personal judgement on you, it's just that that relationship wasn't right. In short, leaving is fairly expected during the early stages and shouldn't be a cause of so much anxiety. He also shouldn't be pushing you to say you'll definitely be with him because that's unrealistic and means that he's not accepting the early stages being kind of like a trial period.
You have a history of abusive relationships
Red potential: It means that you should be more alert to potential signs since you know there's a tendency that something about abusive relationships draws you in.
Benign potential: Obviously your history doesn't say anything about him personally.
Saying nothing nice about his ex
Red potential: He has a tendency to categorise women into good/bad which means he'll be likely to put you on some unattainable pedestal and then rage when you inevitably fall off it. He accepts no responsibility for what might have gone wrong in his previous relationship which means he's unlikely to take responsibility for anything which might go wrong with yours.
Benign potential: Maybe his ex really was that bad. Of course women can be abusive, too. It's possible that his account of things is accurate.
I think that for me the biggest red flag in your situation is this extreme anxiety over you leaving and cheating. It's not something he should be putting on you this early. But even without that, it's about looking at all of the flagged points as a pattern of behaviour. There's always a benign explanation for each red flag individually, but the more red flags you have, the more likely it is that they are linked and the red explanation is the correct one, not the benign one. Learning to join the dots is a useful skill in spotting abusive relationship dynamics.