Does the time of Hishtachavah apply only to the obligation to bring a sacrifice, but not to lashes?

Does the time of Hishtachavah apply if Reuven intentionally became Tamei in the Mikdash?

If he left on the shortest path, he is exempt, but only if he ran.

If he ran out on a longer path, taking the amount of time as walking normally on the shortest path he is exempt.

Hastening the Avodah is considered Avodah.

One enters a house with Tzara'as walking backwards, even if he is entirely inside except for his nose, he is not Tamei.