2. the sum of zero terms (the empty sum) is 0; the product of zero terms (the empty product) is 1
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3. the degree of the zero polynomial is -∞
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4. the maximum of zero numbers is -∞; the minimum of zero numbers is +∞
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5. the determinant of the 0x0 matrix (the empty matrix, with no entries) is 1
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6. the empty graph is disconnected, and so is the empty topological space, for the same reason that 1 isn't a prime number
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7. there's a different empty manifold in each dimension: a zero-dimensional empty manifold, a 1-dimensional empty manifold, etc. (and of course none of them are connected or path-connected)
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nah, if you want to assign the empty space a topological dimension it should be -∞ just like the degree of the empty polynomial. among other things this maintains additivity under product
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not familiar with the inductive dimension. according to wikipedia the inductive definition of the empty set is -1 and i guess you need that to make the induction to points work out. there is a statement along these lines i'll defend, which is that the empty set is the (-1)-sphere
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having the (-1)-sphere be empty extends a bunch of patterns in the spheres down to (-1) correctly, among other things: the euler characteristic of S^n is 1 + (-1)^n, and S^n is n-connective and (n-1)-connected