Batter Hit By Throw while running out of three foot zone.
 

Two Situations, MLB rules
1.
One out, runners on 1B and 3B, Squeeze Play, batter successfully bunts ball but is hit by throw while running out of 3 foot zone and called out. Runner on 3B was halfway to dugout after touching HP when this happened. Runners return to original bases?

2.
Same as above but scored tied in last of 9th.

My understanding is that no runner can advance on Interference.

MLB also talks about an "intervening play" which you did not address in your post. There are some times where a run could count w/ less than 2 out and you could still have INT...but I think you return the runner on the plays your describing.

RTFM

Rule 2.00 (Interference) Comment: In the event the batter-runner has not reached first base, all runners shall return to the base last occupied at the time of the pitch.

Thats my first inclination also, but in last of 9th situation, I might think differently if both runners from 1B and 3B had already touched their next respective bases, which case I would think the game is over and interference ignored.

Why would the B9 be treated any differently from B3 in the situations you are describing?

No run can score if the 3rd out is made at 1B before the B/R reaches 1B.

Rich posted the rule and there are many, many case plays that support the rule.

With one out, if the runner on 1st reached 2nd, the only out that can be made is the 2nd out on the batter. Since a runner on 1B has touched 2B the force situation is off and the run would score even if that runner was eventually put out.

Here are some principles...please cite others if you find them:

If the umpire declares the batter, batter-runner or a runner out for interference, all other runners shall return to the last base that was, in the judgment of the umpire legally touched at the time of the interference unless otherwise provided by the rules.

7.08b comment ..."no runner shall advance on an interference play and a runner is considered to occupy a base until he legally has reached the next succeeding base"

J/R page 109: If a batter/runner has not yet touched or passed first base at the time of interference, all runners not out must return to their TOP base.

Exceptions (the one that matters here) is the intervening play.

Also read OBR rule 2.00 "Interference"

The play is the same regardless of the outs or the inning. BR is out, other runners return (okay -- if the BR was the third out, the runners don't literally return, but they do for scoring purposes).

Only if there's an intervening play would R3's run count.

Bottom 9, one out R1 & R3. batter hit with throw to 1B. 2outs now still R1 & R3. Next batter hits grounder to F4, easy out at 1B - Top 10, tie score - lead off goes yard. Next 3 batters out 1-2-3. Home team goes 3 up-3 down. Visitors win in 10.

Not enforcing the rule changed the outcome. Why do you allow home team to win because it's B9?

We are there to enforce the rules. Rule says Batter out, runners return (as played in OP).

no

Intervening Play?
 

straight from rule 2.00--

On any interference the ball is dead.

Therefore, no play can follow the interference.

FALSE FALSE FALSE. CI and BI are two examples where plays follow interference.

Intervening play is when we have a play at the plate (or any other base), runner safe, and then a INT on any subsequent runner where his INT is 3rd out. Most likely R3 2 outs tapped back to pitcher who flips home, runner safe, then batter/runner INT on play at first. Score the run, batter out. That is the intervening play.

The way the J/R explains it, the intervening play applies only to "a runner from 3B [who] has acquired home plate despite a play against him." [Italics in original.]

It follows, "[T]he run is allowed to score, unless the INT was a third out."

Flip what I said two posts above. Just re-read my own words.

Less than 2 outs, if play is made on runner, and he is safe, this is the base he is returned to when enforcing INT.

If 2 outs, same ruling, just can't have runs score if 3rd out is on B/R before he reaches first or on a force play elsewhere in the infield.

Most of the time this will happen home to first but theoretically it could happen at any base. Just not plausible.

I was going to ask this question in another thread but I'll just add it on here.
I gave a presentation at a winter baseball clinic recently on the running lane. I used alot of the material in Carl Childress(my personal umpiring hero) 2004
book The Usual Suspects. That material talks about the running lane being created in 1882 and that at the time the foul line ran directly through the middle of both first & third base.
Two questions for someone who has the knowledge:
1) When was the foul line(actually fair line) moved to the edge of the base?
2) When they did that did they compensate for the 90 degree angle at home
and move second to keep the diamond a perfect square?