"My blouse is nowhere to be seen."
が kinds of makes the blouse the subject, while は gives it a context of "as for the blouse, ...". With は, you were already kind of talking about the blouse, while が suddenly throws attention on it specifically, most likely without prior conversation about the blouse. You wouldn't really say "As for the blouse, it is nowhere to be found" It's weird that you were already talking about the blouse and then suddenly said that it's nowhere to be found. A few strange contexts could be made I guess, but が makes more sense.
That being said, は should be acceptable as well.
I guess that your example should be acceptable too.
Yes, ありません is the polite form of ない. Sometimes, people simply prefer to use the construction of negative plain form + です when they want to be polite, because words with -ません rather have some certainty and firmness.
I suppose that は should be accepted as well instead of が because the sentence is negative. However, it would sound weird using は unless you haven't established it earlier in the conversation.
And in the end... yes, the に in どこにも is needed because you are describing the location.
is the 私の at the beginning really necessary? I thought if you just said "blouse" it would be assumed to belong to the speaker...