After much frustration and running in circles, I managed to get this to work. I don't know if this is going to help EVERYONE, but perhaps it will warm the souls of many of you who are pulling their teeth out trying to get submenu(s) to work with Mobile Joomla. I am going to start with a fresh installation of the module in my instructions. If you have already installed it, skip to the step where you find yourself!

This outline is for administrators with the following DESKTOP mod_mainmenu structure (the number of parent/child menu items is not important):

Parent - 1
Child - 1-1
Child - 1-2
Parent - 2
Child - 2-1
Parent - 3
Child - 3-1
Child - 3-2
Child - 3-3

1) Goto Extensions>Install/Uninstall
2) Upload the Mobile Joomla Installation Package
3) Once installed, go to Menus>Menu Manager
4)Click New
4a) For Unique Name, enter the name of "Parent - 1" from your site (example: mobileparent1)
4b) For Title, enter the title to be displayed in the Menus dropdown (example: Mobile Parent 1 Menu)
4c) For Description, enter whatever you want (example: Mobile Parent 1 Submenu)
4d) For Module Title, enter the title of the menu module to be created (example: Mobile Parent 1 Submenu)
5) Click Save
6) Goto Menu>Main Menu
7) Put a check in the boxes next to Child - 1-1 and Child - 1-2, and click Copy
Click on the menu you just created (example: mobileparent1) and click Copy
9) Click Extensions>Module Manager
10) Open the module you created when creating the menu (example: Mobile Parent 1 Submenu)
11) Enable the module and deactivate the title, unless you want it displayed
12) Change Position to mj_iphone_middle2 (I found that "middle" displayed NOTHING, but 'middle2' works)
13) Under Menu Assignment, change it to Select Menu Item(s) from the List
14) Select the following menu items:
mainmenu: Parent - 1
mainmenu: Child - 1-1
mainmenu: Child - 1-2
mobileparent1: Child - 1-1
mobileparent1: Child - 1-2

NOTE: You have to select the items from two different menus because if you select the parent from the main/toplevel menu, it will display the article connected to the parent + the mobileparent1 menu (submenu items). If you, then, select something from the mobileparent1 menu, it will open that article, so you STILL want the submenu to appear when the succeeding selections within that submenu.

15) Save your module.
16) Repeat steps 3-16 for each set of CHILD menu items that you want displayed from each parent item.

And hopefully, to answer your final question... do you have to do this for imode, iphone, pda, and wap? The answer to that question is YES. You have to create a submenu for each type of emulation or the submenu(s) won't appear on that type of device.

I hope this helps EVERYONE.

Tell me this is not true...