For the first question, let's be more careful(with the same notations). Let be an open subscheme which factors through for some . So defines a section of named , we want to show its restriction is equal to . Note that has an etale cover , So we only need to check each piece . We look at the stalks and choose a geometric point .

On the one hand, the stalk of restriction of is just the stalk of at pt, which is considered as a geometric point of .

On the other hand, the stalk of is just the stalk of at pt, which is considered as a geometric point of .

Now the key point is we only know pt factors through . But by definition and coincide at some scheme which is etale over such that and both factor through . So can be strictly "smaller" than .

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