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Sebastian got 1 and 3 right. For number 2, here is the definition from the REP you reference, with my comments
However, the pose of a robot in the odom frame is guaranteed to be continuous, meaning that the pose of a mobile platform in the odom frame always evolves in a smooth way, without discrete jumps.
Continuity here is not meant in the mathmatical distinction between discrete and continuous. It means, that you always have small errors, but no jumps in position. You can assume that the difference between the current and previous position estimates is approximately the distance travelled.
The map frame is not continuous, meaning the pose of a mobile platform in the map frame can change in discrete jumps at any time.
If you revisit a place, your robot (its SLAM software) might recognize that place and its position estimate could change very much instantaneously. Therefore you shouldn't assume that the difference between the current and previous position estimates is the actual distance travelled.
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