You're failing to recognize the semicolon. "and for any Speech or Debate in either House, they shall not be questioned in any other Place" is a separate clause from the first part of the sentence, not a clarification of the scope.
Gravel v. United States[1], a Supreme Court case from the 1970s related to Senator Mike Gravel reading the Pentagon Papers (classified material) into the Senate record, established this clearly:
"Rather, his insistence is that the Speech or Debate Clause at the very least protects him from criminal or civil liability and from questioning elsewhere than in the Senate, with respect to the events occurring at the subcommittee hearing at which the Pentagon Papers were introduced into the public record. To us this claim is incontrovertible.
[...]
We have no doubt that Senator Gravel may not be made to answer —either in terms of questions or in terms of defending himself from prosecution—for the events that occurred at the subcommittee meeting."
With regard to this:
> Would a Senator be safe in discussing the classified details in the House?
"House of Congress" and "House of Representatives" are two different things. "the House" generally refers to the latter. "Their House" refers to the House of Congress the member actually belongs to.
The clause is at least ambiguous on the question of whether a Senator has immunity in the House of Representatives, and whether a Representative has immunity in the Senate, but in this case it's unimportant. My hope is that a Senator will walk on to the floor of the Senate, or a Representative will walk on to the floor of the House of Representatives.
[1]http://scholar.google.com/scholar_case?case=2739100783028836...