In the current Official Baseball Rules (pdf), there is only one reference to a "fourth out" that I can find. Page 49, part of Rule 5.09(c) - Making an Out, says:
Appeal plays may require an umpire to recognize an apparent
“fourth out.” If the third out is made during a play in which an appeal
play is sustained on another runner, the appeal play decision takes
precedence in determining the out. If there is more than one appeal
during a play that ends a half-inning, the defense may elect to take
the out that gives it the advantage.
In the case an appeal is made on a force out (i.e. the batter-runner missed first base) and the appeal is upheld, the appeal takes precedence and the run would not count.
This case is a little more complicated. Given that the batter-runner may stop or slow running because of the tag on the player running from 2nd to 3rd, I would guess that the force out would not apply and the run would count. I am having some trouble finding definitive documentation on that. But, Rule 5.09e - Retiring The Side says:
When three offensive players are legally put out, that team takes the
field and the opposing team becomes the offensive team.
It seems to me like the third out in your scenario would end the inning immediately and the batter-runner would no longer be required to touch first base. Wikipedia disagrees with me, but there is no citation for the specific scenario.
If step 3 doesn't happen, then the run definitely counts. Third outs only negate a run if they are a force or happen before the runner crosses the plate. See rule 5.08a:
(a) One run shall be scored each time a runner legally advances to
and touches first, second, third and home base before three men
are put out to end the inning.
EXCEPTION: A run is not scored if the runner
advances to home base during a play in which the third
out is made (1) by the batter-runner before he touches
first base; (2) by any runner being forced out; or (3) by
a preceding runner who is declared out because he
failed to touch one of the bases.
Since none of the three criteria apply, the run would score.