OK,
To make the iPhone template work again you need to undo what you did. So go to the template manager and select the iPhone template. Set menu assignment to (none) and save it,
and then select it again and set menu assignment to (select from list) but do not choose any menu items. Then save it.
Now to get the menu right I recommend you make a new menu with just the "cats dogs lions" and you do it by going to Menus > Menu Manager
If you have a lot of "animals" that are in your PC menu, you may want to copy the PC menu and delete those you do not need. Otherwise create a new menu "mobilemenu" and put the same references that you use in the PC menu for those animals you want.
OK, now you have 2 menus PC menu (probably called Main Menu) and the mobilemenu.
Go to Extensions > Module Manager
in the filter put Main and click go (I am using the English name here, your Joomla may be in a different language)
You should now see perhaps 5 Main Menu modules. Do not panic. They are the modules which show the menu on the PC (unless that's done by your template) and the 4 types of mobile browser.
Click on the one that has mod_mj_iphone_menu under the column called Type
On the right hand side of the screen you will see Parameters > Module Parameters > Menu name
and it is now a drop down menu. Choose mobilemenu and click save.
You should now get the right menu on the iPhone.
Let me know if it works or not.