TLC explores the state graph in a breadth-first search. When it encounters a state that violates an invariant, it reports the invariant violation and stops. (Temporal) properties are evaluated periodically during the construction of the state graph and again at the end when the entire graph has been computed, so it is not guaranteed that TLC will stop at the earliest possible point, and counter-examples to temporal properties are also not guaranteed to be of minimal length.
We are saying "Every behavior that satisfies Spec has a suffix for which P is true". If P happens to be a state predicate, then this is the same as saying "Every behavior that satisfies Spec will eventually reach a state where P is true".
Yes, if P is a state predicate. In general, it means "Every suffix of any behavior satisfying Spec must satisfy P".
Unlike CTL model checking, LTL / TLA model checking doesn't work by recursively checking the sub-formulas of a formula, and in fact it doesn't make sense to say that a state satisfies a property of the form P since temporal formulas are evaluated over sequences of states, not states. Markus already provided pointers to the literature, the textbooks by Baier/Katoen  and by Clarke, Grumberg et al.  are probably the most comprehensive ones.
If the activation status of processes and the leader (i.e., the result of election) are externally observable and your high-level spec says that election happens immediately when the activation status changes then refinements must also do this. Otherwise, an external observer could detect a difference between the behavior of the low-level system (where first the activation status changes and later a process is elected) and the high-level system (where the two happen simultaneously). For example, you might prove an invariant of the high-level spec that asserts that the leader is always active, and that invariant could be violated in the lower-level spec where the current leader gets deactivated before a new leader is elected. Refinement preserves all properties.
Hope this helps,
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