If I donate a brand new item and give it to a charity unopened still in the original packaging, isn't the value of the new item the same as what I paid for it?

Yes.  Select "Comparable Sales" when entering it.

Not always.  Some items drop in value over time.  For example, if you bought a new PC in 1998 for $1000 and it was still new in the box, it could not be fairly valued as $1000 today.  The donation value of an item is what it is worth on the date it was donated.

Good point, but many donations are made relatively quickly after the purchase,  For example, you go to a store and buy things for an upcoming church rummage sale or charity auction.