Not necessarily. The index fund is supposed to track the index. It doesn't need to actually buy the exact stocks as the index, though in the case of the S&P 500, that might be the easiest strategy.
ETF still have different replication strategies. To the customer they guarantee to track the index, however they might buy completely unrelated stocks or options to achieve this. The replication strategy should be an important factor when buying an ETF.
Yes, and SPY is the monster ETF that it is because there is no funny business and it tracks the index the old fashioned way by actually holding the securities. Broker dealers can trade in a basket of securities for new shares of SPY and vice versa which keeps it very close to perfectly tracking the index.
I agree with what you're saying but I was talking about the SPY example listed above. The prospectus [0] for SPY says that it's not actively managed and that it rebalances at least monthly.
Not that big a bet on BH's part.