The closest investment analogy to a bitcoin hard fork could be a stock split.
No.
If a hard fork occurs, the total number of bitcoin's will increase from 20 million to 40 million coins.
Wrong again. The supply of Bitcoin remains the same. What happens is that you create an altcoin with the same (existing and projection) supply of Bitcoin.
That could mean that bitcoin's price will split by roughly 1/2.
No.