[BoomerSooner]

Bob...I think I actually misread the question and do think he's asking about the batter-runner.

The same basic principles still apply to the batter runner with the added thought process being that at most only 1 out was going to be obtained on the play as described (with normal play, at least 1 runner was still going to be on base). Since the error wasn't the cause of the batter-runner making it to first, he would be an earned run unless he scored on or after a play which would have been the 3rd out without the original error or any subsequent errors.

Examples (to avoid confusion, the original runner on 2nd will be Abel, Baker will be the batter and subsequent players will follow accordingly):

1. The scorer determines that the error by F1 prevented Abel from being put out at 3rd and Abel scores on the play. This run is unearned and results in Baker reaching first base on a Fielder's Choice. Following this play...

A. With no outs and Baker on 1st, Charlie hits a home run.
Baker and Charlie are earned runs as the error did not contribute to them scoring.

B. With no outs and Baker on 1st, Charlie is put out on a sacrifice bunt and Baker advances to 2nd base. With 1 out and Baker on 2nd, Daniel strikes out but Baker steals 3rd during the AB. Now with 2 outs and Baker on 3rd, any sequence of plays (base hit, error, SB, WP, PB, etc) allows Baker to score.
Baker's run is an unearned run as the original error prevented Abel from being put out. Had the error not been made, the inning would have ended on Daniel's strikeout prior to Baker scoring.

2. The scorer determines that the error by F1 did not prevent Abel from being put out at 3rd, but did allow Abel to score on the play. This run is unearned for the moment and the inning will be reconstructed with Abel on 3rd to determine if he subsequent becomes an earned run. Play continues with Baker reaching first base on either a base hit or FC (this is a judgement call that I won't delve into here). Following this play...

A. With no outs and Baker at 1st, Charlie hits a home run.
The runs scored by Abel, Baker and Charlie are earned runs.

B. With no outs and Baker at 1st, Charlie strikes out. With 1 out and Baker at 1st, Daniel strikes out. With 2 outs and Baker at 1st, Edward strikes out.
The run scored by Abel is unearned. Without the error, Abel would have been stranded at 3rd base.

C. With no outs and Baker at 1st, Charlie is put out on a sacrifice bunt and Baker advances to 2nd base. As the rule gives the pitcher the benefit of the doubt in determining what bases runners might have advanced, the theoretical Abel remains on 3rd base. With 1 out and Baker on 2nd, Daniel strikes out. With 2 outs and Baker on 2nd, Edward hits a home run.
The runs scored by Abel, Baker and Edward are all earned runs. The error by F1 was not determined to prevent a put out, so the defense has not had a chance to make the 3rd out prior to the runs scoring.

D. With no outs and Baker at 1st, Charlie reaches 1st base on an error due to a misplayed sacrifice bunt attempt and Baker advances to 2nd (benefit of the doubt to the pitcher leaves theoretical Abel at 3rd). Daniel pops up to shallow left, but F7 drops the ball. Baker advances to 3rd, Charlie advances to 2nd and Daniel reaches on the error. Without the errors, the theoretical situation is 2 outs with Abel on 3rd, Baker on 2nd (it is fair to assume he would have advanced on the sacrifice bunt, but not necessarily so for Abel). The actual situation now has 1 out with Baker on 3rd, Charlie on 2nd and Daniel on 1st. Edward grounds out to first base, Baker scores, Charlie advances to 3rd and Daniel advances to 2nd.
Abel's and Baker's runs are unearned as are any subsequent runs allowed.

E. With no outs and Baker at 1st, Charlie reaches 1st base on an error due to a misplayed sacrifice bunt attempt and Baker advances to 2nd (benefit of the doubt to the pitcher leaves theoretical Abel remains at 3rd). Daniel pops up to shallow left, but F7 drops the ball. Baker advances to 3rd, Charlie advances to 2nd and Daniel reaches on the error. Without the errors, the theoretical situation is 2 outs with Abel on 3rd, Baker on 2nd (it is fair to assume he would have advanced on the sacrifice bunt, but not necessarily so for Abel). The actual situation now has 1 out with Baker on 3rd, Charlie on 2nd and Daniel on 1st. Edward hits a home run scoring Baker, Charlie, Daniel and Edward.
Runs scored by Abel, Baker and Edward are now earned, but runs by Charlie and Daniel are unearned as they reached base due to errors that prevented them from being put out.