How to get the publishing and menu fields from the node edit form in the "Manage fields" tab?
Issue solved mainly thank to the Renderable elements module.

It adds the core node edit form's vertical tabs and their content to the usual Manage fields tab, allowing any kind of customization (adding, removing, reordering tabs and fields, the tabs properties can even be turned into horizontal tabs if one needs this, etc.).

There is a very good short video introducing this module's main features.

The only limitation is that the way this module alter the forms may cause incompatibilities with other modules. On my side I just noticed minor issues with no impact to the end-user:


The standard checkbox "Provide a menu link" does not hide automatically the other menu related options depending on its state, this is a non-issue for me since I precisely wanted to customize this behavior,
Modules like Conditional Fields do not detect yet the supplementary fields added by the Renderable Elements module. The few States API effects directly involving these fields might need to be coded specifically (Conditional Fields offers an option to manually define the jQuery selector to be used, I didn't tried yet if this could be a reliable workaround for this issue),
Other modules which were already programmatically altering the default vertical tabs may be confused and become unable to alter the tabs anymore. For instance, Fast Permissions Administration does not add anymore its "Manage Permissions" link in the Publishing options vertical tab.


The good news is that this module is fully compatible with Display Suite and Field Group. The vertical tabs and its components can be moved to any Display Suite's region, but more importantly the vertical tabs content can be organized in any kind of group you like (&lt;DIV&gt; markers, (collapsible) fieldsets, other tabs, etc.).