Particles always modify what comes before them, like a suffix, so you can't move と before ともだち like it's a prefix and still have those elements working together. The と particle follows a noun to mean that the verb is done "along with" that noun, so it's also not right for it to follow the は particle instead of a noun. (As a general hint, pretty much no particle will ever follow は.)
ともだちえいが is also not a correct phrase, as generally two nouns (ともだち and えいが) cannot directly follow each other without particles following them clarifying their relationship to each other or the verb.
To break down what the particles are doing in each part of the sentence:
カオは → Kao (marked as the topic of conversation with は)
ともだちと → Friend (marked as an additional person doing the verb with と)
えいがを → Movie (marked as what the verb is acting on with を)
みます → To watch (the verb all of the other particle-marked elements are relative to)
Particles work, as stated, like a suffix. It's postpositional, I believe is the term. That means you have to have a phrase, in this case, noun phrase, and attach a particle after it. That is not to say that you can't reposition the と if you need to move it around, but it would be grammatically correct to move the attached word with it. But keep in mind, even though Japanese is incredibly flexible, sometimes it will sound funny, and sometimes there are set phrases that you absolutely can't change that part.
Some particles are also attached to other particles, and they're put in a certain order. So they're special pieces of a puzzle that fit to certain things, depending on their context. So you need to keep in mind that you can't just simply move a particle around when you feel like. They provide information of the function that the word is performing. So above all, it has to belong to the word it is meant for, so as long as you maintain that, you're still comprehensible.
Also, while Japanese seems incredibly dynamic, don't make light of omissions either. Sometimes Japanese people like to condense their speech, and since communication relies on context as a crutch, omissions do not mean a particle and such do not exist. Context upholds their existence. It's just hidden, not shifted around and swept under the rug, or else you will get lost and communication breaks down.
