Youre only 25, throw this guy back, there are plenty of fish in the sea.  You can already see what your life will be with him: 2 undisciplined kids, a problem BM.  I can almost guarantee BM will be dumping them off more and more while she demands more and more money.  Read around on this site if you don't believe me.  Run, girl

You've only been in this relationship 4 months? Cut your losses and get out of this situation.  Your BF is a lousy parent,  his ex sounds high conflict and the kids drive you nuts. Why in the world would you want to stay? Love isn't always enough.  Save yourself. 

Chalk this up to experience and move on. Why shackle yourself to a guy who doesn't care about your feelings enough to keep his kids in tow? This will not get better.

You don't need this and the kids don't need this. They do just sound like 1 and 2 year olds but not many would want to take on someone else's kids at this age.

How recent was the split? Is this a situation where he moved in with you because he couldn't afford to live on his own? While there are exceptions to the rule, it is not a great idea to get so quickly involved with someone who is just coming out of a divorce. He needs time to figure out how to be a  single father on his own, before he starts living with someone else.

You should not be discipling the kids, their father should do it. To be honest, they sound like typical 1 and 2 year olds. Perhaps the problem is just that you hate kids. There is nothing wrong with that, but you shouldn't be living with a man who has kids, if you hate them.

This does not sound like the relationship for you. You are young, find a man who does not have kids, and don't move in with him right away.

This isn't going to work for you. You obviously aren't ready for an instant family. You are young, and have only been with this guy for 4 months and are having these feelings. They are not going to go away, and will probably get worse as the kids get older. 

The right thing to do would be to move on.

Leave. That is the only real option here for a multitude of reasons.

First, your BF is a crap father. Independent of you as a person, it's a really crappy parenting move to move your new SO into your house. It's a really crappy parenting move to get that serious that quickly with someone. It's a really bad parenting move to take less than a year to figure out how to be a single parent before jumping in to the next serious relationship. 

Second, your BF is a crappy partner. Moved you in but doesn't want you to have a say about how the kids should act? Allows BM to send hate messages without sending her a cease and desist from an attorney? Basically, he's leaving you WIDE OPEN to deal with his parenting AND relationship failures, probably because you're making his life easier (e.g. splitting rent, having sex with him, likely doing a lot of the parental tasks for him) while it makes your life harder.

Third, BM sounds crazy. Or, she's not crazy and he lied to you about his relationship status with her and you are the unwitting "homewrecker". This isn't your fault, unless you knew he was in a relationship when you pursued him, in which case that's partially on you. But, at minimum, there is another mandatory 17 YEARS where BM can get crazier if she truly is crazy, and you're cracking at 4 months.

Which brings me to the fourth and final - you're not currently cut out to be a SM. Many of the behaviors you described by the kids are normal for their age. What they need is a strong parent who corrects their bad behaviors. I don't know from what you've said if your BF does that but it's still not good enough for you, or if he truly is a crap parent who just lets them be brats. Either way, you hate them. They make your life worse. It's not fair to you or them that they're hated because they either behave like babies/toddlers or they are poorly parented.

This isn't the relationship for you, or anyone. Your BF needs to learn how to be a single parent before he ties himself to someone. He needs to put a figurative muzzle on BM, especially to anyone he dates. If his 2 year old has autism, then your BF needs to focus on setting up therapy and structure for that child. NONE of that has been accomplished, and that alone should make him not eligible to date, much less jump into a serious relationship with.

Remember, LOVE ISN'T ENOUGH. Victims of domestic violence still love their abusers. That doesn't mean these are good relationships that need/should continue.

Love is a really crappy measure by which to gauge the health and sustainability of a relationship. Most folks who end relationships don't end them because there was cheating or violence or some big, grand, awful episode. It's death by a thousand cuts that typically have nothing to do with love. Most relationships end with the people in that relationship still being in love, but they are no longer (or never were) compatible.

Let this one go. At 25 you can find a man with no kids and be much happier. You have love blinders on now, but this guy and all of his baggage won't seem so wonderful in another year or so. 

Did he move into your place or did you move into his? If he moved in to your place and isn't trying to teach his kids to behave, I'd ask him to leave, find his own place and work on being with his kids. This is especially apprpriate seeing as he is apparently going to get them full time (?). If you moved into his place, my bags would already be packed. The fact that he decided to take them so much without even discussing it with you would have been the last straw for me.

Get out while you can! I'm sure you love him, but the pros don't outweigh the cons. You still have a whole life ahead of you... so you really wants infants and toddlers around in your 20s? NO THANKS. 

I am piling on here.  4 months? 25 years old?  Time to leave.. it's not fair to anyone for you to stay in this relationship.. not him.. not his kids.. not yourself.  It just isn't for you.