Yes. It should be Peachy State.
yes
You capitalize the "s" for state when you referring to a specific state or state agency; State of Texas or State of Maine or The State Dept. of Health. You don't capitalize the "s" when the word state is used in general terms; the states with the largest population or the western most state.
Yes, "State" should be capitalized in "State law" when referring to laws specific to a particular state within a country.
It should be capitalized if you are referring to a specific state government but not when you're referring to state governments in general.
Yes, "State" should be capitalized in "State of Iowa" because it refers to the political entity of Iowa as a state within the United States.
Capitalize the word state only when it appears after a state's name, as in "We will travel to Washington State this summer." (But make sure it won't be confused with a visit to the university known as Washington State. Perhaps "state of Washington" would be better.) In the phrase "state of Hawaii," you don't capitalize the word state. Finally, don't capitalize the word state when it's being used as a substitute for the state's name, as in "My father works for the state." It is capitalized, however, in imaginative names such as "the Nutmeg State," "the Empire State," "the Aloha State," and we capitalize "States" when we say things like "We're returning to the States after twenty years in Europe."
If it is the word 'state' you capitalize the first 's' like this 'State'.
Minnesota is the Gopher State.
Yes, always capitalize a state.
When it is used as a name of the theater, then it should be capitalized.
Wisconsin is the 30th state. Minnesota is the 32nd state.