It seems that bottom gap can be eliminated rather easily by defining some heights in Application.Resources.
App.xaml
<Application.Resources>
<x:Double x:Key="AppBarThemeMinHeight">56</x:Double>
<x:Double x:Key="AppBarThemeCompactHeight">40</x:Double>
</Application.Resources>
MainPage.xaml
<Page.BottomAppBar>
<CommandBar>
<CommandBar.PrimaryCommands>
<AppBarButton Label="New" >
<AppBarButton.Icon>
<FontIcon Glyph="" FontSize="16"/>
</AppBarButton.Icon>
</AppBarButton>
<AppBarButton Label="Select">
<AppBarButton.Icon>
<FontIcon Glyph="" FontSize="16"/>
</AppBarButton.Icon>
</AppBarButton>
</CommandBar.PrimaryCommands>
</CommandBar>
</Page.BottomAppBar>
Fine Alignment (Optional)
AppBarButton (and/or AppBarToggleButton)
To adjust icon position of each button, put a copy of the default AppBarButton style in Application.Resources and alter Margin of ContentePresenter(Name="Content").
<ContentPresenter x:Name="Content" Margin="0,10,0,3" ... />
CommandBar
To adjust ellipsis button's position, similarly make a copy of CommandBar style and alter Padding of Button(Name="MoreButton").
<Button x:Name="MoreButton" Padding="16,17,16,0" ... />
Then,

All the explation here-in-above are based on the default styles defined in generic.xaml ver 10.0.14393.