I've created my own component and I can't find the file with the settings for the title name in the backend. Whenever the current view is my component it displays the name properly but if the component is another one it just shows com-mycomponent in the toolbar as the name. Also I can't figure out how to add a dropdown like the banners component does.
I think you are taking about component menu in the administrator section.
For this you need to create a system language file which will contain all the component menu translation.
In the component xml file you need to add entry of this language file like shown in the code
<languages> <language tag="en-GB">languages/admin/en-GB.com_component.ini</language> <language tag="en-GB">languages/admin/en-GB.com_component.sys.ini</language> </languages>
When the current view is of your component then the main language file that is en-GB.com_component.ini gets loaded which might contains the desired language translations.
But in case of different component view the desired translation is missing because by default joomla does not loads language files of all the component on all views.
But it loads system language files on every view.
So you should put component menu translation in this file.