in 1, you should preferably use "est" only once. -> "Case parva, sed pulchra est." In fact, you may drop "est" altogether. When a form of "esse" in the indicative functions as a copula, it is often omitted. -> casa parva, sed pulchra.
in 6 you should make some changes: it should be -> "mea nemo est magistra, me ipsa doceo". Why? The dative of possession is used, not to emphasize to whom something or someone belongs, but to stress the thing or person owned. Yet in this case, the person to whom (doesn't) belong(s) is the focal point: Arachne is indicating "no one is my master, I am my own master", not "I don't have a master (as opposed to something else), etc". The place of the possessive pronoun is free, yet primary position seems sensible, to even more emphasize the negation of possession.
"ipsa" should be added, to intensify the idea of autonomous action.
Add.: to elaborate on "ipsa" -> without ipsa the sentence "me doceo" would simply, casually assert a fact: I teach myself. But, mindful of the context, we can see, that Arachne is not making a neutral assertion of fact. She is contrasting that fact with a contrary idea (that someone else would be her master). Therefore, to express this contrast, she should stress that element, which forms the contrast.