Summarizing my answer from law.SE... According a NYT analysis, the law only bars certain executive functionaries like the inspector general from making such disclosures; senators/legislators are not on that list.
Do whistle-blowers have a right to remain anonymous?
Only in a limited way. Another part of the Inspector General Act says that agency watchdogs “shall not, after receipt of a complaint or information from an employee, disclose the identity of the employee without the consent of the employee, unless the inspector general determines such disclosure is unavoidable.”
In line with that law, the inspector general for the intelligence community, Michael Atkinson, did not include the whistle-blower’s name in his report to the acting director of national intelligence, Joseph Maguire. Mr. Maguire testified last week that he did not know the name of the person [...]
But the legal prohibition on disclosing the official’s name applies only to Mr. Atkinson. It does not bar Mr. Trump and his allies from trying to identify him or disclosing his name if they figure it out. (It would be illegal under the Intelligence Identities Protection Act for any official to disclose his name if he is a covert agent, but no one has suggested that he is.)
(There's a quote from OIG that confirms that on law.SE; I'm omitting it here.)
As for the senate rules on walking out... there's a distinction to be made between the rules of the Senate that have been voted on and the instructions passed out by leadership for the trial.
Nothing in the official Senate impeachment rules forbids it. The rules, dating from March 2, 1868, make little mention of the chamber at all, except to determine that the doors to the Senate should be left open, unless the members are in deliberations.
Senate leaders Mitch McConnell (R-KY) and Chuck Schumer (D-NY) circulated their own guidance ahead of the trial, where they made passing mention of the attendance expectations.
“Senators should plan to be in attendance at all times during the proceedings,” they wrote in the decorum guidelines.
“During the impeachment proceedings, standing will not be permitted on the floor and this requirement will be strictly enforced,” they also wrote. “Accordingly, all Senators are requested to remain in their seats at all times they are on the Senate floor during the impeachment proceedings.”
But that’s about it. And some senators are having little trouble bucking what limited guidance they’ve been given.
Per CNN, Sen. Lindsey Graham (R-SC) spent more time out of his seat than in it Wednesday night. Sen. Dianne Feinstein (D-CA) also left for stretches, while Sen. Cory Booker (D-NJ) could be seen using his iPhone in the Democratic cloakroom (electronics are banned from the chamber during the trial).
I.e. the guidance on being in attendance at all times has been shirked more broadly.