Let's say a batter hits a foul popup to the catcher, and it's an easy play but the catcher drops it. Later in the at-bat, the batter gets a solid hit. Is this scored as the batter reaching on an error, or getting a hit? What about any runs that are created that would not otherwise have scored? Are these earned or unearned?
Your question is a bit unclear..
Later in the at-bat, the batter gets a solid hit. Is this scored as the batter reaching on an error, or getting a hit?
Well, when the batter gets a solid hit, does anybody playing the field make an error, or was it a clean hit?
The pop-up to the catcher has nothing to do with the solid hit. If the catcher drops the ball, that is an error on the catcher for that 1 play, but that play doesn't have any influence on the judgement of a future play as far as how the batter gets on base
Reaching on an Error means that the batter reached base safely on the current play due to a fielder making an error during that play, not from a past play.
As far as earned runs charged to a pitcher in this situation
Rule 9.06 b) No run shall be earned when scored by a runner who reaches first base (1) on a hit or otherwise after his time at bat is prolonged by a muffed foul fly;
Just to answer the remaining questions as well and clarify any further confusion, you can't have an error on a foul ball. A dropped foul ball is just a foul ball. Anything happening after that plays out as it does. Errors only count if the hitter gets on base as a direct result of it (i.e. pops it up and the catcher drops it in fair territory and the batter reaches 1st because of it).
The best way to figure out unearned runs is to replay the inning without the errors. For instance, if batter 1 strikes out, batter 2 pops out to RF, batter 3 grounds it to SS and reaches because of a bad throw to 1st (an error on the SS), and then batter 4 hits a homerun.... replaying that inning without errors gives you a strike out, fly out, ground out (would have been if the throw were accurate). Therefore, the two runs scoring on the homerun were unearned. The hit still counts against the pitcher, so his final line for that inning is 1 IP, 1 H, 2 R, 0 ER, 0 BB, 1 K.
I hope that helps.
Given my interpretation of the first question I will pose it in my words. First pitch the batter hits a foul pop up that the fielder drops that should have been caught. That fielder gets an error. The next pitch the batter hits a clean single. For that at bat it is my opinion the batter gets a hit so his box score shows a single even though an error is also present. Basis 10.12 indicates the drop foul fall prolonged the at bat. Thus, to that point there was no at bat charged.