I'm going to present to you a few different schools of thought for this situation. I'm not sure which one is going to be most appropriate to you, perhaps a blending of them will be necessary.
You've indicated that you are 13, but have not indicated the ages of your siblings. Based on the chores you've listed, I'm guessing around the 5-7 range. Furthermore, I'm guessing it's probably summer break.
There's no way that you should be responsible for the care of your siblings. They are your father's children and his responsibility, not yours.
You're not your siblings' mother, nor should anyone be asking you to be.
You are more than capable of this
The reality of the situation is that your father needs to work for most of the day and you are of an age where you can take on some responsibility of your own. To that end, you can also help out with taking care of your siblings.
It is likely that your father is trying to teach you these skills by more or less pushing the experiences onto you and is trying to encourage you to rise to the challenge.
I strongly suspect your father is likely feeling overwhelmed at the amount of work he's trying to juggle as a single father, primary household income earner, household administrator (i.e. signing kids up for school, maintaining the insurance, paying the bills, identifying and making repairs), and general housework (vacuuming, laundry, dishes, trash, etc.).
As a coping method, he's pushing the parenting work onto you in part because you are able to do it. And I think some of it you should at least engage in, but preferably following a conversation with your father regarding where your responsibilities begin and end.
With my own daughter (6 years at the time of writing), I have made her responsible for a few chores around the house, in part because she is capable of doing the work; in part because her doing the work genuinely helps the family; and in part because I want her to understand that the entire household needs everyone to contribute in order to work. To exemplify, those chores currently are feeding and refilling the water for our cats; it is irrelevant that one of the cats is older and definitely mine, just as it's irrelevant that one of them is younger and definitely hers. What is relevant is that feeding the cats is her job, her doing the job helps the house, and that she is capable of doing that job on her own.
I think the issue of the dog business that you'd mentioned is a good example of this. As I'm teaching my daughter this lesson regarding responsibility, I find it is important to be consistent in my expectations. Thus, when I realize the cat food is low, I do not fill it myself. I want my daughter to fill it on her own and not learn to expect me to do that job for her and I will remind her of her job the next time I see her. I strongly suspect that is what's occurring regarding the dog's business. Your dad stepped in it because you hadn't taken let the dog out, so now he's making your clean up the mess, which while perhaps harsh is likely a fair consequence.
Now let's focus on the issue of your siblings. First of all, I will be clear that technically none of the jobs you listed are your responsibility by default. However, the life situation whereby your dad is a single parent working most of the daylight hours and is probably regularly exhausted does create a practical issue whereby he is likely seeking help from wherever he can. The chores you listed sound like a lot to ask if it's all the time, but also some of the time they are likely within your ability to manage well enough.
I think given the nature of the situation you describe, it behooves you to talk with your dad directly about how this is making you feel. The timing of this conversation, if possible, should strive to be for a day when he's in a good mood if you can; specifically, it's important that he has sufficient energy to give it proper thought and attention.
I wouldn't expect things to necessarily get better right away, but simply expressing your feelings to your father is going to make him aware that you're feeling overwhelmed by what he's asking of you and hopefully he's able to find alternative means to divert some of the load. Perhaps the other parents can take the kids more often, perhaps your dad takes on more of the parenting work but needs your help with more of the household work (i.e. vacuuming, laundry, dusting, dishes, etc.), perhaps your dad starts sending the siblings to daycare.
Ultimately, nothing can happen until you speak with him about it.