FCC regulates them in the US, "Submarine Cable License", they don't just pop out of any document system though. The one I did find:

http://licensing.fcc.gov/myibfs/download.do?attachment_key=9...

Uses Google data to show the locations, along with the lat/long (using the textual lat longs should be fine from a copyright/data license perspective, though I think lots of OSM people would hem and haw about it being in the same document as Google data).

A lot of the filings are ownership updates to existing cables and don't contain any information about the locations of the cables.

It's not open; but you can get cable locations near shore in marine maps.

Most cable cut incidents occur either fishing or anchoring a vessel. That's why cable operators send this data to cartography providers.

In the UK and Europe it largely is. KIS-ORCA is one project.

http://www.kis-orca.eu/map