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For (1), consider the case of $V = \mathbb{P}^1$, the projective line. The only regular functions are the constants, which are obviously not dense in the space of all smooth $\mathbb{C}$-valued functions on the $2$-sphere.

Added: For that matter, consider the affine line $\mathbb{A}^1$. I think to make your question more reasonable, you should also throw in complex conjugates of the regular functions (c.f. Hodge theory).

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For (1), consider the case of $V = \mathbb{P}^1$, the projective line.

Added: For that matter, consider the affine line $\mathbb{A}^1$. I think to make your question more reasonable, you should also throw in complex conjugates of the regular functions (c.f. Hodge theory).

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For (1), consider the case of $V = \mathbb{P}^1$, the projective line.