Do the prices change during after-market hours? (Or do they remain independent?)
Yes, prices change overnight. A great reference on the issue is Lou, Polk and Spouras (2018).
They basically argue that there are investors who prefer to trade overnight whereas other prefer to trade during the day:
For example, and as the primary focus in our analysis, some investors may prefer to trade at or near the morning open while others may prefer to trade during the rest of the day up to and including the market close. Since these two periods when the market is open vs. when it is closed differ [?] along several key dimensions, including information on [?], price impact, and borrowing costs, it seems likely that many aspects of investor heterogeneity that might be relevant for asset pricing also manifest as a tendency to trade in one of these periods rather than the other. In this light, the presence of overnight and ìntraday clienteles seems a reasonable and perhaps even natural starting point.
Indeed it also seems that many known trading strategies earn most of their returns overnight rather than when markets are open. Take a look at their table II:
In short. Yes. the prices do change, but not all symbols trade, volume and liquidity is generally much lower with wider spreads. I would personally DE-emphasize the price action unless there is trading after release of earnings-- where volume is significant as there is more noise with larger spreads even on liquid issuers a relatively small buyer/seller can move the market after hours.
The opening session is driven more by futures markets if single issue futures are available or algos trading correlations with index futures-- but ultimately the closing price has nothing to do with where the open is. The open is created by the pre-open limit order levels and whoever hits the bid or ask with a market order first makes the print for the open.