Hey lesanjo, this may be of some use to you and to others who may be struggling with it. I ended up getting the separate mobile menu working in Joomla 3. Here is my setup and how I did it:
I'm using my own responsive template as a separate mobile template with MobileJoomla. It shares all of the same module positions as my desktop template.
1. You don't have to copy any menu. I think a lot of people are getting confused on this part, but it's really not a necessary step. Simply create a new menu and name it what you want. (ie: Site-Mobile-Menu).
2. Once you've created the menu, create a module for it (I suggest naming the module the same as the menu), and assign the module to the same position as the menu you want to replace. (In my case, I'm replacing the Main Menu located in position-1). Save and Close.
3. Still in Module Manager, open Mobile Menu (located in position mj_all_header). Under options you'll see it's assigned to Main Menu by default. Change that to the name of the new menu you just created.
4. Open up MobileJoomla component, make sure you've selected your correct template for each device, click on Advanced Settings, then click on Modules. You'll see the list of all modules, find the module for your new menu and deactivate it for Desktop, leave activated for all mobile devices. Now find the module for your desktop menu (if you did it right it should be in the same position as your new menu), deactivate the module for all mobile devices and leave activated for Desktop only. Save your changes.
5. Add any menu items you like into your new menu. If you have items that will point to the same pages as your desktop site, make sure to use Menu Item Alias under System Links to point to the correct page.
6. Check out your site on mobile and you should now see the correct menu.
Now, as for Forcing Homepage, you are right about that. It doesn't work correctly for me either. It will load the correct homepage that I direct it too, but for whatever reason it locks and redirects all links to the homepage. Not good. As a work-around, I'm using the Mobile Content plugin on my default desktop homepage (which is currently doubling as my mobile homepage), and simply wrapping {desktop} and {mobile} tags around the appropriate content.
I hope that I've helped with this info. Oh and BTW, I tried responding to this earlier from Firefox, but this site is displaying the mobile version in Firefox desktop? Weird. I'm now using Chrome to write this and all is well.
Take care.