How can I emotionally present when my kids have big emotions I’m not comfortable with?

I love this question, because I can definitely sense that you care about improving the relationships with your kids, as well as growing in your own mental health journey. One of the best parts of this, is that as you learn and grow, so do they. Kids are like sponges, waiting for the next thing to soak up, and consistently demonstrating a skill can help them latch on to it. The brain and nervous system are trainable. Right now, it sounds like there are some responses that are not working. The responses that we give, and the responses that we get, can create a feedback loop. For example, if we walk into a situation already feeling frustrated, thinking that it will not go well -> the situation does not go well -> kid becomes frustrated and expects that the same outcome will happen again the next time. So the next time said situation happens, both people are frustrated and respond accordingly. Each situation has an opportunity for a teaching moment. A teaching moment requires us to think about how we want to handle it *before* it happens again, so that we are prepared to go into it in a more intentional way, and get a new response. 

Part one of this is going to involve reflecting on what you want to see in these relationships. What comes to mind first: setting healthy boundaries, personal space, naming emotions, and allowing them to see your emotions in a healthy way. Part two: learning to manage your emotions so that you are able to act as a support for them. They are watching and learning constantly, so the more managed we can be in our nervous system, the more managed theirs will be, too so they will be able to think and act more clearly. Part three: Look at their behavior as a means to get their needs met. Think about your own emotions and needs. If you are feeling tired, the need is "get sleep or rest". When a child hits or bites, what is the need? What is the "why" behind it? Attention, affection, communication, to manage emotions (yes, they could be trying to manage this way and need to see alternatives from you). Once we can see behavior and emotion as an unmet need, our views will change and needs can be met - which leads to improved relationships.