Can we edit Mobile App Salesforce Navigation through code / callouts to Metadata?

Hi,

Yes,you can  edit Mobile App Salesforce Navigation through code / callouts to Metadata. It should be under "AppMenu" metadata ("Salesforce1.appMenu"). You should be able to pull the changes and push the changes through metadata with the following structure:

<AppMenu xmlns="http://soap.sforce.com/2006/04/metadata">
  <appMenuItems>
    <name>MyDay</name>
    <type>StandardAppMenuItem</type>
  </appMenuItems>
  <appMenuItems>
    <name>Tasks</name>
    <type>StandardAppMenuItem</type>
  </appMenuItems>
  <appMenuItems>
    <name>Feed</name>
    <type>StandardAppMenuItem</type>
  </appMenuItems>
  <appMenuItems>
    <name>Dashboards</name>
    <type>StandardAppMenuItem</type>
  </appMenuItems>
  <appMenuItems>
    <name>Search</name>
    <type>StandardAppMenuItem</type>
  </appMenuItems>
  <appMenuItems>
    <name>Events</name>
    <type>StandardAppMenuItem</type>
  </appMenuItems>
  <appMenuItems>
    <name>DE_Widget</name>
    <type>CustomTab</type>
  </appMenuItems>
</AppMenu>