Your partner has a previous commitment to her three children, which must supercede her relationship with you. You have to accept that. Also, you are not a parent to them. You are an adult friend, in a similar position as a teacher or a coach or a favourite uncle, who is also a housemate and their equal in that respect, if you have moved into their house. And that "friend" distinction is not given automatically, but must be earned.
If you have problems with neatness or decorum, as kurro said, talk with them. You can also talk with them and their mother, raising the issues you have and suggesting rules for the household. I'm not sure what the comment about cutting meat off the bone means, but if the entire family is more comfortable with clutter than you are, then maybe you will have to adapt. You can't force your values on them. Certainly if you move in and try to tell them they can't run around their own house you won't be too popular. Respect is a two-way street. The only thing you can do is give respect, for your part. The rest is up to them. And yet I find that kids are most often willing to give respect.
It is possible that the fact that you are out of work has influenced your mood and made you less good company than you are wont to be. If that is the case, that is a separate issue you need to address. You can't be taking out your frustrations on your housemates, whatever their ages. You say that these kids, and all kids, seem to have the ability to wind you up. Part of being the adult is not to be wound up. Your partner needs another adult, she doesn't need you to be a fifth child she needs to adjudicate squabbles among.
Find out what the kids expect of you, and how you aren't measuring up. Find you what your partner expects of you with respect to her three older kids, and how you are not measuring up. Let them know you are going to make an effort to change, and ask for their help letting you. Then change. Respect from the kids will follow.