While more can always be said (especially since this only addresses one of multiple questions), you may be interested in one of my previous answers in which I discuss some fines given to children in cotton mills, like "Riding on each other's back", "Making a noise when order'not", and "Terrifying S. Pearson with her ugly face".

I can't speak to explicit examples of children's actions in those various places and to be sure, I'll defer to those who study what happened in those particular spaces, but I can shed some light on why those details are likely hard to come by.

To put it plainly, children in that era had limited agency and few, if any, opportunities to document their own lives or create enduring documents. So, their appearance in the historical record depends on the degree to which the adults around them saw their actions as worthy of writing down. Charlotte Hardman, one of the first anthropologists of childhood, wrote in 1971 that the history of children (and women) is "muted." Children and women were, she said, "unperceived or elusive groups (in terms of anyone studying a society)." Hardmen contributed to a field of study known as the sociology of childhood which incorporates history and anthropology into its work and offers a paradigm for thinking about childhood. The relevant features of the paradigm that apply to our understanding of children in history are (from James & Prout, 1997):

1. Childhood is understood as a social construction. As such it provides an interpretive frame for contextualizing the early years of human life. Childhood, as distinct from biological immaturity, is neither a natural nor universal feature of human groups but appears as a specific structural and cultural component of many societies.
2. Childhood is a variable of social analysis. It can never be entirely divorced from other variables such as class, gender, or ethnicity. Comparative and cross-cultural analysis reveals a variety of childhoods rather than a single and universal phenomenon.
3. Children’s social relationships and cultures are worthy of study in their own right, independent of the perspective and concerns of adults.
4. Children are and must be seen as active in the construction and determination of their own social lives, the lives of those around them and of the societies in which they live. Children are not just the passive subjects of social structures and processes.
5. Ethnography is a particularly useful methodology for the study of childhood. It allows children a more direct voice and participation in the production of sociological data than is usually possible through experimental or survey styles of research.

(I get into ethnography and how that relates to children in history in my answer to the question, "What's the history behind asking children, "what is your favorite color?")

What this all means in terms of your question is that when adults do mention children in those settings, they're usually doing it in service to their own goals, rather than neutrally describing the actions of new-ish humans. This same phenomenon appears in writing about schools or education. I can rattle off anecdotes I've come across from teacher journals from the mid-1800s where a child is described as destroying books but she writes home that the same child was an absolute hellion to her but through sheer force of will and motherly instincts, she was able to calm him down and teach him his letters. And of by the way, if the school board was interested in buying new books for the school, she knows this particular boy simply loves to read. Which is to say, adults can be unreliable narrator with regards to the things children in her charge did. So, if you asked how did children behaved in school in the early 1800s, I could only tell you how adults described their behavior or how they thought specific behaviors should be approached. We can, and do, make inferences but they're usually followed by lots of caveats or very narrowly focused. Take for example, /u/jbdyer's response where we have a limited explanation of what the workers, which included children, was exactly doing. In other words, one person's "dancing in room" is another person's moving their body because they're tired or in pain.

None of this is to say adults didn't recognize children were different - millions of words were written in the 1800s about what should be done with children, how they should be parented, etc. etc. However, the idea that children are just children is, unfortunately, a lot more complicated than it sounds. A historical detail that can help us get our heads around how complicated it is the idea of a teenager didn't take shape until the 1950s and 1960s. This doesn't mean teenagers didn't exist before that moment, but rather, the idea that there is something so specific about being aged 13-18, it deserves its own name emerged as high school attendance was normalized. Before teenager, the concept of adolescence itself was created by adults. Even toddler is a concept that didn't always exist. (More on that here in an older answer about medieval toddlers and refusing food.)

I feel comfortable saying a something similar to what I said in that post: did children who worked in factories, mills, and mines strongly and vocally express their opinions about work? Probably. Did adults see such opinions as abnormal behavior that needed to be handled? Likely not. And the reason for this is because the notion that a strong, vocal reaction from a child is something that needs to be handled or is abnormal behavior didn't exist in that era as we think of it today. From my older response:

The very notion of a tantrum is a fairly modern - 20th century - concept. That is, the idea of toddler expressing their opinion strongly and vocally in response to a request for an adult deserves its own name ("tantrum") emerged from the idea that there is "typical" child behavior - or normal and abnormal behavior that needs to be redressed.

I wrote about the field of Child Study in response to the question, "What is the history behind "What is your favorite color?"" and the movement's impact can be seen in your question, and in fact, the curiosity behind your question. In effect, they asked the same thing you're asking: why do small humans do these things that big humans don't? The founder of the movement, a man named G. Stanley Hall, was fascinated by this idea that the attributes of childhood could be observed and studied in the same way scientists studied the natural world. This isn't to say adults before the Hall came along in the late 1800s weren't curious about children's motivations - or had opinions - but rather, Hall created structures that gave rise to developmental psychology, child psychiatry, and societies for Mental Hygiene focused on children's behavior.

While adults in previous eras would and could describe children's behavior on a continuum or scale in relation to their siblings or other children, what Hall and his contemporaries did was related to scale and norming. They collected thousands of anecdotes about children, detailing everything toddlers did and wrote about patterns. The field of child psychology encouraged doctors and parents to frame children's behaviors as good or bad, healthy or unhealthy. Kathleen Jones 1999 book, Taming the troublesome child: American families, child guidance, and the limits of psychiatric authority goes into more detail about how this history evolved and explores how the theories behind a child's "tantrum" ran the gauntlet from "their mother gave them too much attention as a baby" to "their mother didn't give them enough attention as a baby." Public health, especially child health, was a very popular social issue in the early 1900s and doctors around the warned of disastrous events if a child's bad behavior weren't fixed. Thus, tantrums became something to be handled and fixed, rather than something children did.