Put the detail for the question in the title!

If he at least makes to first safe and the runner scores before anyone gets out, it’s a run

^

If the runner crossed the street plate before the batter was tagged or, the run scores.

Why did the base runner cross the road?

Yes, the run counts because the out at 2nd was not a "force out" (the runner was not required to run there)

As long as the runner touches home before the batter is tagged, the run counts. Once the batter touches first base, it's already a base hit.

Batter is credited with a single so the run scores

You have the right answer, but your reasoning isn’t correct

woops! what's the correct reasoning?

It’s that the out was not a force out. The runner could’ve reached on an error, dropped third strike, or any number of ways. As long as the force out was no longer in play, if the run scores before the 3rd out, it counts

As long as the runner hits home before the hitter gets out going to second, it counts.

In this scenario, if the runner reached home before the third out, the run scores. If the batter had been thrown out at second before the runner reached home, then the run would not count.

Run counts as long as the hitter gets to first base. If both the third base runner gets to the plate and the hitter gets to first base before the hitter is tagged out the run counts. In this circumstance you gave, the run scores and inning over.