How can I show my newly created app on the Salesforce1 mobile app? It doesn't appear as a menu option with Mobile Navigation. Also, how can I remove objects that are currently being shown?
By "Apps", they don't actually mean a collection of tabs (e.g. under "Create > Apps"), but rather a single Visualforce page that is "mobile enabled" that has a Visualforce tab. If you create a page that meets these qualifications, you can use it in Salesforce1. Note that this page can navigate to other visualforce pages, but the app shows only the first page.
Might be you would like to use Flexipage.
If you want to see an "app" in mobile you will need to add it into Navigation Menu. Go to Mobile Administration --> Mobile Navigation.
But please Have a look to the thread below to get more info about Flexipages and App concept in SF1 .
Take a look at the documentation.
Salesforce Desktop Apps and Apps section in the SF1 Nav are not equivalent.
Apps section in the SF1 Nav are counter-intuitively equivalent to Salesforce Desktop Tabs.
The link above shows you have to add Custom Tabs to the SF1 Nav menu. If the custom tab is a VF page, it must be enabled for mobile (see other responses).
A key caveat is that the first 4/5 tabs in the Selected column (in this documentation) will show up almost as pinned tabs in the nav menu, while the other tabs will be put under the Apps section of the menu.