Your DB looks in good health.
Here is what I have done on a fresh WP 3.7.1 with no plugin activated except Polylang.
Test with Polylang 1.1.6 -> every looks OK
Upgrade to Polylang 1.2.3 -> No language as you currently experience
De-activate Polylang 1.2.3 (no need to delete anything).
Activate: "Revert Polylang 1.2 to 1.1.6" http://pastebin.com/0TdQzQLv
De-activate: "Revert Polylang 1.2 to 1.1.6"
Activate Polylang 1.2.3 -> everything looks OK
I noticed: no menu locations and no widgets (since I don't have your theme). All strings translations are in French (but it is how they are in the original db). One string translated in English is in DB (correctly upgraded to 1.2.3) and does not appear in strings translations table (probably used by a widget or a plugin which I did not activate).
I don't understand what could be wrong before you updated. Probably the only useful thing in "Revert Polylang 1.2 to 1.1.6" is getting back the version number to 1.1.6 as I did not see any other data from Polylang 1.2.3 in your database.