Well, here are 3 I'm pretty sure of. I'll see if I manage to look at the rest, but I need more time
1. False. First off, the product cannot be written like you did. You're taking the product of the elements of the bases, not of the bases themselves. That aside, you need to restrict the product to only finitely many indices (the rest being the whole space).
2. A countable (need not be finite) product of second countable spaces is second countable, and every second countable space is separable.
4. Isn't it always true, independently of being finite or not? If you pick an open neighbourhood of in , the inverse image of is open since is continuous, and it obviously contains .