Cross-file repetitions are just those segments which are repetitions (they have the same content) of the first instance of a certain segment that is placed in the primary file (the file in which a given segment occurs for the first time). In other words, if you have segments in other files with are repetitions of a certain segment from the primary file, those are called cross-file repetitions.

The analysis of a project takes place in a "top-to-bottom" fashion, so to say. When you create a project with multiple files that have the same segment, the first instance of it will be marked as Unmatched. Other instances of that segment in other files will be marked as a Repetition with the information: Repeat - matched across files:

If a linguist sees such information, it just means that this segment was not translated yet in any file.

This is more information than the actual repetition. If any instance of that segment in any file is translated, then other occurrences will receive the relevant translation if a linguist accesses them in Workbench in the "Edit" mode:

Please, be advised that repetitions and cross-file repetitions are two separate functionalities.

The repetitions update is based on the user's settings in Workbench. 

The cross-file repetitions, however, are subject to the normal update of matches, i.e. the current segment + 2 segments below will be updated.

That being said, if a given cross-file repetition receives a TM match upon the project analysis, its first instance will have a ICE/Leveraged status whereas its instances in subsequent files will receive Leveraged status, and as such they will be displayed in Project Metrics (external matches always take precedence over any kind of internal matching). This is caused by the fact, that it was not marked as a repetition during the analysis of the file:

Although there cannot be a match from the exact same segment where it has been translated, without reanalysis, it is expected that the first occurrence can receive matching from the other segment, in this case, the child repetition.

Will such a segment, which is matched to another file that has been removed from the project, be counted as Fuzzy Match or as No Matching?

That depends which file will be removed. If you have two files in a project - one with an unmatched segment and the second one with information: Repeat - matched across files, and you delete the file where the segment is unmatched, the second file will unfortunately have a false repetition unless you re-analyze the whole project. In general, when files are added/deleted after the project creation, a full project re-analysis is required in order to calculate cross-file repetitions properly.

The "hide repeated segments" option

XTM allows for enabling the option to hide repeated segments, and this also applies to cross-file repetitions (of course, except for the first instance of the segment). The said setting can be enabled in Configuration -> Settings -> Translation -> TM -> Repeats:

Once enabled, the option is activated in the "create a new project" form when you can specify the percentage of a given file that repetitions occupy at which the option shall be actually activated:

Please, remember that while repeated segments in Workbench can be hidden, they are automatically (and silently) populated with the newest translation in subsequent files, thus they will be translated accordingly in a target file once the latter is re-generated. Also, the Metrics will still display those repetitions, and the file concerned will show the original number of segments in the Project editor -> Workflow section.