I agree with Thumbwitch - this does sound appalling, not only the mother's attitude but also your DP's IMO.
Unless the mother has genuine reason for you not seeing the children - e.g. if you were a violent criminal, a heroin addict or a raging alcoholic - she has no right to insist you don't. The children are also your DP's and he should be capable of deciding what people, places and experiences are appropriate for them to see and do. Presumably, he has no power of veto over anyone she introduces them to ?
Okay .... so with someone this unreasonable, I totally "get" why he's reluctant to rock the boat as contact is used as a punishment when he "transgresses" and unfortunately many NRPs with unreasonable exes are controlled in exactly this way, with an ever present threat of no contact being used by the ex to make demands which aren't fair. However, he should at least be trying to resolve this - out of respect for you (you're not some "dirty secret" and also because he should have an honest relationship with his kids where they're included in all appropriate aspects of his life - including the family life he now has with you .... which (shock horror) might actually be a positive experience for them). If the ex messes with contact I think he should obtain a contact order meaning, in theory at least, that she can't just pull contact on a whim (and if she does she'll be in breach of court). This is relatively easy to do and pay for if you self rep and the ex would have to come up with very good reasons in court for preventing you seeing the kids - it's likely that if she tried she'd end up looking like a complete fool.
All the while he does nothing he's not only being disrespectful to you (IMO) but also potentially compounding the issue for the future. If contact is used as a threat what else might she demand and/or try to control this way ? My feeling is that if this sort of thing isn't nipped in the bud as soon as possible you end up living in limbo and not really being in control of your own life because you never really know how the ex is going to act in any given situation where she doesn't get her own way. As Thumbwitch also said, what'll happen if you have a child together ? It's crazy that your DP is compartmentalising his life and not a healthy situation for him or his kids. Whether the ex likes it or not he has a life with you now - and effectively denying that when he sees his kids is ridiculous. No-one's suggesting you must always be around when he sees them but for them never - presumably - to set foot in their father's home and participate in normal family life with him is a farce.
With all that in mind, whilst I don't think it's right you say "horrible things" about his kids, I do also think it's perfectly understandable for you to say they mean nothing to you - you don't know them ! I hope your DP isn't pressuring you to assume some sort of false "feeling" for them, that's just silly in the circumstances and he needs to step up and sort this out for your sake and for their theirs. His too really, he's letting himself be controlled by a (spiteful ? vengeful ?) ex for no good reason. And personally, I find it extremely unattractive to have a partner kow-tow to an ex because they're frightened of her - it really affects the respect you have for them when they'd rather upset you and/or accept something very unfair, than stand up to her.
Also, why, if you send anything to his email, is she able to access that ? Think it's about time he changed his password(s) surely ? And setting the police on you sounds more like she's doing the harassing - I'm very surprised they'd have responded to such a complaint, has she made other accusations ? To protect yourselves make sure you record every communication with her, just in case this "harassment" thing escalates.