Runner enters dugout

Question
Theres a runner on 3rd base and the batter hits a foul ball but the umpire doesnt call foul ball, so the runner at 3rd thinks it a pass ball she scores and goes in the dung out. Then the umpire says it a foul ball. He puts the runner back on 3rd base. The other team says she should be out because she went into dead ball area. I would think the runner can go back because its a dead ball. Is that the correct call?

Response
The runner should be placed back on third base.
If a runner abandons their base and enters the dugout, then they are out. (8.18.Y)
However based on your description, that rule does not apply.

3 Foot Lane Violation

Question
I have a question on a thrown ball from the catcher who just fielded a bunt throws to first base, the batter baserunner has not reached the 3ft running lane the thrown ball hits the runner in the back and the runner is on the line or just inside fair terr, is the runner out? Same question but the ball is on a dropped 3rd strike or a pass ball is the runner out?

Response
The batter-runner is not out unless the act was intentional. Since the batter-runner was less than half way done the line 8-17-E does not apply.
Note case book play under Three Foot Running Lane
SITUATION A: With the bases loaded, B4, who bats left-handed, hits a ground ball to F3, who is left-handed. The ball is just inside the foul line. F3 throws to F2, but hits B4, who is on her way to first base. B4 is in fair territory, but not reached the 30-foot running lane.
RULING: There is no violation, unless the batter-runner intentionally interfered with F3’s throw. (8-17-E, 8-18-F)