(If you write a put and a call for the same strike, you are basically in the same position as a short seller. If you buy a put and a call for the same strike, you are economically in the same position as an owner of the stock.)
This is not entirely accurate. It is the same position as borrowing some amount of money to buy the stock. If I buy a put and call and the price at expiry is exactly that strike I am guaranteed to lose money.
Not quite. It's the 'risk-free bond' term that I dropped from my explanation of the put call parity.
To replicate the stock, you'd buy the call and sell the put. The risk exactly balances out in the sense that a total portfolio of 1 stock short, 1 call long and 1 put short would have zero risk and behave like a risk-free bond.
(If the risk premia of the long call and the short put would not exactly balance, you could make money with very simple arbitrage trades.)
Economically, sure, but not literally. If you hold both a put and a call, for instance, you can't vote with them, or earn dividends. You would not say you're the actual owner -- whereas a short sale creates a negative actual stock.
(If you write a put and a call for the same strike, you are basically in the same position as a short seller. If you buy a put and a call for the same strike, you are economically in the same position as an owner of the stock.)
Hence, you can't separate options from stocks.